The document discusses matrices and determinants. It defines a matrix as a rectangular table with numbers or formulas as entries. It provides examples of 2x2 and 3x3 matrices. The document explains that square matrices have a number called the determinant extracted from them. It then discusses how the 2x2 determinant represents the signed area of a parallelogram defined by the row vectors, and explores properties of the sign of the determinant. Finally, it suggests generalizing these concepts to 3x3 determinants.
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M210-S16-class-5.pdf
M 210 More drawing in LATEX February 23, 2016
Arcs
An arc of radius r from point (p,q) with initial angle ϕ (phi) and final angle ψ (psi) is drawn in
TikZ by the code
LATEX\draw (p,q) arc (phi:psi:r);
Note that TikZ does not require the center (a,b), which is often easier to determine that the arc’s
initial point. Polar coordinates (relative to the arc’s center) can be used to express the initial point
in terms of the arc’s center.
(a,b)
(p,q)
ϕ
ψ
r
p = a + r cosϕ
q = b + r sinϕ
In tikzpicture environment enter
\draw ({a+r*cos(phi)},{b+r*sin(phi)}) arc (phi:psi:r);
for the specific values of a, b, r, ϕ (phi) and ψ (psi).
LATEX
According to the TikZ’s manual, by default arcs are drawn in positive (counter-clockwise) direction,
assuming initial angle ϕ to be smaller than ψ. However, I have been able to draw arcs in negative
(clockwise) direction by choosing an initial angle larger than final angle.
Precise Label Positioning
The relative positions above, below, left, right, above left, above right, below left, and
below right, provide positioning relative to the node (indicated by the dot) as illustrated in the
following figures.
below
above
left right
above left
below left
above right
below right
While it is possible to adjust the separation between label and node, it is not that easy to adjust
the angle (the above provide label placement only for angles in multiples of 45◦). The following will
work to place the label (that is its center) at the the indicated position: precise distance s at relative
angle θ from the position of the node.
node
lab
el
s
node coordinates (nx,ny)
label coordinates (lx, ly)
θ
To place label at the above mentioned position use the code (where θ is theta degrees):
M 210 — February 23, 2016 page 44
LATEX\draw ({nx + s*cos(theta)}, {ny + s*sin(theta)}) node {label};
To rotate the label as in the above illustration, use code:
LATEX\draw ({nx + s*cos(theta)}, {ny + s*sin(theta)}) node {\rotatebox{theta}{label}};
Special Triangles
In trigonometry the triangles with degrees 45-45-90 and 30-60-90 provide exact values for the trigono-
metric functions. It is quite easy to obtain the exact values of angles of 15 and 75 degrees from those
of 30 degrees by half- and double-angle formulas and other trigonometric identities, however, these
values can also be obtained from the following figure containing an equilateral triangle in a square
with sides equal to 2.
1
The Regular Pentagon. Use trigonometric functions to draw a regular pentagon with its lower
side along the line segment from (−1,0) to (1,0). Start by drawing this line segment.
LATEX\begin{center}
\begin{tikzpicture}[scale=2]
\draw[line width=1.2pt] (-1,0) -- (1,0);
\end{tikzpicture}
\end{center}
Extend the line segment on both sides by adding the following code.
LATEX\draw[line width=1.2pt] (1,0) -- ({1+2*cos(72)},{2*sin(72)});
\draw[line width=1.2pt] ...
Concept of Particles and Free Body Diagram
Why FBD diagrams are used during the analysis?
It enables us to check the body for equilibrium.
By considering the FBD, we can clearly define the exact system of forces which we must use in the investigation of any constrained body.
It helps to identify the forces and ensures the correct use of equation of equilibrium.
Note:
Reactions on two contacting bodies are equal and opposite on account of Newton's III Law.
The type of reactions produced depends on the nature of contact between the bodies as well as that of the surfaces.
Sometimes it is necessary to consider internal free bodies such that the contacting surfaces lie within the given body. Such a free body needs to be analyzed when the body is deformable.
Physical Meaning of Equilibrium and its essence in Structural Application
The state of rest (in appropriate inertial frame) of a system particles and/or rigid bodies is called equilibrium.
A particle is said to be in equilibrium if it is in rest. A rigid body is said to be in equilibrium if the constituent particles contained on it are in equilibrium.
The rigid body in equilibrium means the body is stable.
Equilibrium means net force and net moment acting on the body is zero.
Essence in Structural Engineering
To find the unknown parameters such as reaction forces and moments induced by the body.
In Structural Engineering, the major problem is to identify the external reactions, internal forces and stresses on the body which are produced during the loading. For the identification of such parameters, we should assume a body in equilibrium. This assumption provides the necessary equations to determine the unknown parameters.
For the equilibrium body, the number of unknown parameters must be equal to number of available parameters provided by static equilibrium condition.
What is the distance between the points B and C Experience Tradition/tutorial...pinck3124
FOR MORE CLASSES VISIT
www.tutorialoutlet.com
This paper is part of an examination of the College counting towards the award of a degree.
Examinations are governed by the College Regulations under the authority of the Academic Board.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
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2. Determinant and Cross Product
A matrix is a rectangular table with numbers
or formulas as entries.
3. Determinant and Cross Product
A matrix is a rectangular table with numbers
or formulas as entries. The size of a matrix is r x c
where r is the number of rows and c is the number of
columns.
4. Determinant and Cross Product
A matrix is a rectangular table with numbers
or formulas as entries. The size of a matrix is r x c
where r is the number of rows and c is the number of
columns. Following are examples of matrices:
5 2 x 2 –1
A2X2 = 5 2 –3
4 –1 B2X3 = C3X3 = 6 –2 y2
4 –1 0
11 9 –4
5. Determinant and Cross Product
A matrix is a rectangular table with numbers
or formulas as entries. The size of a matrix is r x c
where r is the number of rows and c is the number of
columns. Following are examples of matrices:
5 2 x 2 –1
A2X2 = 5 2 –3
4 –1 B2X3 = C3X3 = 6 –2 y2
4 –1 0
11 9 –4
Matrices of sizes n x n, such as A and C above,
are called square matrices.
6. Determinant and Cross Product
A matrix is a rectangular table with numbers
or formulas as entries. The size of a matrix is r x c
where r is the number of rows and c is the number of
columns. Following are examples of matrices:
5 2 x 2 –1
A2X2 = 5 2 –3
4 –1 B2X3 = C3X3 = 6 –2 y2
4 –1 0
11 9 –4
Matrices of sizes n x n, such as A and C above,
are called square matrices. For each square matrix A,
we extract a number called the determinant of A.
7. Determinant and Cross Product
A matrix is a rectangular table with numbers
or formulas as entries. The size of a matrix is r x c
where r is the number of rows and c is the number of
columns. Following are examples of matrices:
5 2 x 2 –1
A2X2 = 5 2 –3
4 –1 B2X3 = C3X3 = 6 –2 y2
4 –1 0
11 9 –4
Matrices of sizes n x n, such as A and C above,
are called square matrices. For each square matrix A,
we extract a number called the determinant of A.
Given a 2x2 matrix A the determinant of A is
a b
det(A) = det = ad – bc. Hence det(A) = –13.
c d
8. Determinant and Cross Product
The 2x2 determinant
det a b = ad – bc
c d
gives the signed area of the
parallelogram defined by the
row–vectors (a, b) and (c, d),
9. Determinant and Cross Product
The 2x2 determinant
det a b = ad – bc
c d (c, d)
gives the signed area of the
(a, b)
parallelogram defined by the
row–vectors (a, b) and (c, d),
10. Determinant and Cross Product
(a+c,
The 2x2 determinant b+d)
det a b = ad – bc
c d (c, d)
gives the signed area of the
(a, b)
parallelogram defined by the
row–vectors (a, b) and (c, d),
11. Determinant and Cross Product
(a+c,
The 2x2 determinant b+d)
det a b = ad – bc
c d (c, d)
gives the signed area of the
(a, b)
parallelogram defined by the
row–vectors (a, b) and (c, d),
i.e. |det(A)| = the area of
12. Determinant and Cross Product
(a+c,
The 2x2 determinant b+d)
det a b = ad – bc
c d (c, d)
gives the signed area of the
(a, b)
parallelogram defined by the
row–vectors (a, b) and (c, d),
i.e. |det(A)| = the area of
To see this, we dissect the
rectangle that boxes–in the
parallelogram into regions as
shown
13. Determinant and Cross Product
(a+c,
The 2x2 determinant b+d)
det a b = ad – bc
c d (c, d)
gives the signed area of the
(a, b)
parallelogram defined by the
row–vectors (a, b) and (c, d),
i.e. |det(A)| = the area of
(a+c,
To see this, we dissect the b+d)
rectangle that boxes–in the
parallelogram into regions as (c, d)
shown (a, b)
14. Determinant and Cross Product
(a+c,
The 2x2 determinant b+d)
det a b = ad – bc
c d (c, d)
gives the signed area of the
(a, b)
parallelogram defined by the
row–vectors (a, b) and (c, d),
i.e. |det(A)| = the area of
(a+c,
To see this, we dissect the b+d)
rectangle that boxes–in the
parallelogram into regions as (c, d)
shown (a, b)
15. Determinant and Cross Product
(a+c,
The 2x2 determinant b+d)
det a b = ad – bc
c d (c, d)
gives the signed area of the
(a, b)
parallelogram defined by the
row–vectors (a, b) and (c, d),
i.e. |det(A)| = the area of
(a+c,
To see this, we dissect the .
b+d)
rectangle that boxes–in the
parallelogram into regions as (c, d)
shown (a, b)
.
16. Determinant and Cross Product
(a+c,
The 2x2 determinant b+d)
det a b = ad – bc
c d (c, d)
gives the signed area of the
(a, b)
parallelogram defined by the
row–vectors (a, b) and (c, d),
i.e. |det(A)| = the area of
(a+c,
To see this, we dissect the .
b+d)
rectangle that boxes–in the
parallelogram into regions as (c, d)
shown and note that area of (a, b)
.
= – 2( + + )
.
.
17. Determinant and Cross Product
(a+c,
The 2x2 determinant b+d)
det a b = ad – bc
c d (c, d)
gives the signed area of the
(a, b)
parallelogram defined by the
row–vectors (a, b) and (c, d),
i.e. |det(A)| = the area of
(a+c,
To see this, we dissect the .
b+d)
rectangle that boxes–in the
parallelogram into regions as (c, d)
shown and note that area of (a, b)
.
= – 2( + + )
.
.
= ad – bc (check this.)
18. Determinant and Cross Product
The sign of the 2x2 determinant D of the matrix
D = det a b = ad – bc
c d
gives the following information concerning directions.
19. Determinant and Cross Product
The sign of the 2x2 determinant D of the matrix
D = det a b = ad – bc
c d
gives the following information concerning directions.
i. If D > 0, then the parallelogram D = ad – bc > 0
is formed by sweeping u = < a, b> (c, d)
in a counterclockwise motion, (a, b)
i.e. to the left of u. u
20. Determinant and Cross Product
The sign of the 2x2 determinant D of the matrix
D = det a b = ad – bc
c d
gives the following information concerning directions.
i. If D > 0, then the parallelogram D = ad – bc > 0
is formed by sweeping u = < a, b> (c, d)
in a counterclockwise motion, (a, b)
i.e. to the left of u. u
ii. If D < 0, then the parallelogram
is formed by sweeping u = < a, b> D = ad – bc < 0
in a clockwise motion, (a, b)
i.e. to the right of u.
u (c, d)
21. Determinant and Cross Product
The sign of the 2x2 determinant D of the matrix
D = det a b = ad – bc
c d
gives the following information concerning directions.
i. If D > 0, then the parallelogram D = ad – bc > 0
is formed by sweeping u = < a, b> (c, d)
in a counterclockwise motion, (a, b)
i.e. to the left of u. u
ii. If D < 0, then the parallelogram
is formed by sweeping u = < a, b> D = ad – bc < 0
in a clockwise motion, (a, b)
i.e. to the right of u.
u (c, d)
iii. If D = 0, there is no parallelogram.
It’s deformed to a line.
22. Determinant and Cross Product
Example A.
det 1 0 = +1, so from i = < 1, 0> to j = <0, 1> is a
0 1
counterclockwise sweep.
23. Determinant and Cross Product
Example A.
det 1 0 = +1, so from i = < 1, 0> to j = <0, 1> is a
0 1
counterclockwise sweep.
det 0 1 = –1, so from j = <0, 1> to i = <1, 0> is a
1 0
is a clockwise sweep.
24. Determinant and Cross Product
Example A.
det 1 0 = +1, so from i = < 1, 0> to j = <0, 1> is a
0 1
counterclockwise sweep.
det 0 1 = –1, so from j = <0, 1> to i = <1, 0> is a
1 0
is a clockwise sweep.
a b
Hence we say the matrix
c d
preserves orientation if D > 0,
25. Determinant and Cross Product
Example A.
det 1 0 = +1, so from i = < 1, 0> to j = <0, 1> is a
0 1
counterclockwise sweep.
det 0 1 = –1, so from j = <0, 1> to i = <1, 0> is a
1 0
is a clockwise sweep.
a b
Hence we say the matrix
c d
preserves orientation if D > 0, i.e. facing in the direction
of <a, b>, the vector <c, d> is to the left, just as in our
R2 if we face the x–axis, the y–axis is to the left,
26. Determinant and Cross Product
Example A.
det 1 0 = +1, so from i = < 1, 0> to j = <0, 1> is a
0 1
counterclockwise sweep.
det 0 1 = –1, so from j = <0, 1> to i = <1, 0> is a
1 0
is a clockwise sweep.
a b
Hence we say the matrix
c d
preserves orientation if D > 0, i.e. facing in the direction
of <a, b>, the vector <c, d> is to the left, just as in our
R2 if we face the x–axis, the y–axis is to the left,
reverses orientation if D < 0,
27. Determinant and Cross Product
Example A.
det 1 0 = +1, so from i = < 1, 0> to j = <0, 1> is a
0 1
counterclockwise sweep.
det 0 1 = –1, so from j = <0, 1> to i = <1, 0> is a
1 0
is a clockwise sweep.
a b
Hence we say the matrix
c d
preserves orientation if D > 0, i.e. facing in the direction
of <a, b>, the vector <c, d> is to the left, just as in our
R2 if we face the x–axis, the y–axis is to the left,
reverses orientation if D < 0, i.e. facing in the
direction of <a, b>, <c, d> is to the right which is the
mirror image to our choice of R2.
28. Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
29. Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
a b c
det d e f
g h i
30. Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
a b c a b c a b
det d e f d e f d e
g h i
g h i g h
31. Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
a b c a b c a b
det d e f d e f d e
g h i
g h i g h
32. Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
a b c a b c a b
det d e f d e f d e
g h i
g h i g h
33. Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
a b c a b c a b
det d e f d e f d e
g h i
= aei g h i g h
34. Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
+ +
a b c a b c a b
det d e f d e f d e
g h i
= aei + bfg g h i g h
35. Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
+ + +
a b c a b c a b
det d e f d e f d e
g h i
= aei + bfg + cdh g h i g h
36. Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
+ + + –
a b c a b c a b
det d e f d e f d e
g h i
= aei + bfg + cdh – (ceg i g h
g h
37. Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
+ + + – – –
a b c a b c a b
det d e f d e f d e
g h i
= aei + bfg + cdh – (ceg + afhg bdi)
g h i + h
38. Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
+ + + – – –
a b c a b c a b
det d e f d e f d e
g h i
= aei + bfg + cdh – (ceg + afhg bdi)
g h i + h
1 2 –1 1 2 –1 1 2
Hence det 6 –2 2 6 –2 2 6 –2
3 1 –4 3 1 –4 3 1
39. Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
+ + + – – –
a b c a b c a b
det d e f d e f d e
g h i
= aei + bfg + cdh – (ceg + afhg bdi)
g h i + h
8 12 –6
1 2 –1 1 2 –1 1 2
Hence det 6 –2 2 6 –2 2 6 –2
3 1 –4 3 1 –4 3 1
40. Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
+ + + – – –
a b c a b c a b
det d e f d e f d e
g h i
= aei + bfg + cdh – (ceg + afhg bdi)
g h i + h
8 12 –6 6 2 –48
1 2 –1 1 2 –1 1 2
Hence det 6 –2 2 6 –2 2 6 –2
3 1 –4 3 1 –4 3 1
41. Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
+ + + – – –
a b c a b c a b
det d e f d e f d e
g h i
= aei + bfg + cdh – (ceg + afhg bdi)
g h i + h
8 12 –6 6 2 –48
1 2 –1 1 2 –1 1 2
Hence det 6 –2 2 6 –2 2 6 –2
3 1 –4 3 1 –4 3 1
= 8 + 12 – 6 – (6 + 2 – 48)
= 54
42. Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
+ + + – – –
a b c a b c a b
det d e f d e f d e
g h i
= aei + bfg + cdh – (ceg + afhg bdi)
g h i + h
8 12 –6 6 2 –48
1 2 –1 1 2 –1 1 2
Hence det 6 –2 2 6 –2 2 6 –2
3 1 –4 3 1 –4 3 1
= 8 + 12 – 6 – (6 + 2 – 48)
= 54
Following are some of the important geometric and
algebraic properties concerning determinants.
43. Determinant and Cross Product
The geometric significance of the
3x3 determinants is the same as
the 2x2 determinants.
44. Determinant and Cross Product
The geometric significance of the
3x3 determinants is the same as
the 2x2 determinants. z +
<g, h, i>
a b c <d, e, f>
If det d e f = D, <a, b, c>
g h i y
D<0
then D is the signed volume of x
the parallelepiped, i.e. a squashed box,
formed by the row–vectors of the matrix.
45. Determinant and Cross Product
The geometric significance of the
3x3 determinants is the same as
the 2x2 determinants. z +
<g, h, i>
a b c <d, e, f>
If det d e f = D, <a, b, c>
g h i y
D<0
then D is the signed volume of x
the parallelepiped, i.e. a squashed box,
formed by the row–vectors of the matrix.
46. Determinant and Cross Product
The geometric significance of the
3x3 determinants is the same as
the 2x2 determinants. z +
<g, h, i>
a b c <d, e, f>
If det d e f = D, <a, b, c>
g h i y
D<0
then D is the signed volume of x
the parallelepiped, i.e. a squashed box,
formed by the row–vectors of the matrix.
i. If D > 0, then the row vectors
<a, b, c> → <d, e, f>→ <g, h, i> form a right handed
system as in the case i → j → k.
47. Determinant and Cross Product
The geometric significance of the
3x3 determinants is the same as
the 2x2 determinants. z +
<g, h, i>
a b c <d, e, f>
If det d e f = D, <a, b, c>
g h i y
D<0
then D is the signed volume of x
the parallelepiped, i.e. a squashed box,
formed by the row–vectors of the matrix.
i. If D > 0, then the row vectors
<a, b, c> → <d, e, f>→ <g, h, i> form a right handed
system as in the case i → j → k.
ii. If D < 0, then <a, b, c> → <d, e, f>→ <g, h, i>
form a left handed system as in the case i → j → –k.
49. Determinant and Cross Product
Here are some important properties of determinants.
x * *
1 0 0
i. det 0 y * = xyz, hence det 0 1 0 = 1
0 0 z
0 0 1
50. Determinant and Cross Product
Here are some important properties of determinants.
x * *
1 0 0
i. det 0 y * = xyz, hence det 0 1 0 = 1
0 0 z
0 0 1
a b c
ii. Given that det d e f = D, then
g h i
ka kb kc a b c a b c
det d e f = det kd ke kf = det d e f = kD
g h i g h i kg kh ki
where k is a constant,
51. Determinant and Cross Product
Here are some important properties of determinants.
x * *
1 0 0
i. det 0 y * = xyz, hence det 0 1 0 = 1
0 0 z
0 0 1
a b c
ii. Given that det d e f = D, then
g h i
ka kb kc a b c a b c
det d e f = det kd ke kf = det d e f = kD
g h i g h i kg kh ki
where k is a constant, i.e. stretching an edge of the
box by a factor k, then the volume of the box is
changed by the factor k.
52. Determinant and Cross Product
3. The determinant changes sign if we swap two rows
in the matrix.
53. Determinant and Cross Product
3. The determinant changes sign if we swap two rows
in the matrix. For example,
a b c d e f
if det d e f = D, then det a b c = –D.
g h i g h i
54. Determinant and Cross Product
3. The determinant changes sign if we swap two rows
in the matrix. For example,
a b c d e f
if det d e f = D, then det a b c = –D.
g h i g h i
Geometrically, this says that if we switch the order of
any two vector, we change the orientation of the
system defined by the row vectors.
55. Determinant and Cross Product
3. The determinant changes sign if we swap two rows
in the matrix. For example,
a b c d e f
if det d e f = D, then det a b c = –D.
g h i g h i
Geometrically, this says that if we switch the order of
any two vector, we change the orientation of the
system defined by the row vectors.
1 0 0 1 0 0
So det 0 1 0 = 1 and det 0 0 1 = –1
0 0 1 0 1 0
since we changed from a right handed system to a
left handed system following the row vectors.
56. Determinant and Cross Product
In fact the sign of the determinant is the algebraic
definition of the right handed system vs. the left
handed system, i.e. a positive determinant means the
row vectors form a right handed system and a negative
determinant means they form a left handed system.
57. Determinant and Cross Product
In fact the sign of the determinant is the algebraic
definition of the right handed system vs. the left
handed system, i.e. a positive determinant means the
row vectors form a right handed system and a negative
determinant means they form a left handed system.
Cross Product
58. Determinant and Cross Product
In fact the sign of the determinant is the algebraic
definition of the right handed system vs. the left
handed system, i.e. a positive determinant means the
row vectors form a right handed system and a negative
determinant means they form a left handed system.
Cross Product
Given two 3D vectors u = <a, b, c>, v = <d, e, f>,
their cross products are defined below
59. Determinant and Cross Product
In fact the sign of the determinant is the algebraic
definition of the right handed system vs. the left
handed system, i.e. a positive determinant means the
row vectors form a right handed system and a negative
determinant means they form a left handed system.
Cross Product
Given two 3D vectors u = <a, b, c>, v = <d, e, f>,
their cross products are defined below
i j k
u x v = det a b c
d e f
60. Determinant and Cross Product
In fact the sign of the determinant is the algebraic
definition of the right handed system vs. the left
handed system, i.e. a positive determinant means the
row vectors form a right handed system and a negative
determinant means they form a left handed system.
Cross Product
Given two 3D vectors u = <a, b, c>, v = <d, e, f>,
their cross products are defined below
i j k i j k
u x v = det a b c and v x u = det d e f
d e f a b c
61. Determinant and Cross Product
In fact the sign of the determinant is the algebraic
definition of the right handed system vs. the left
handed system, i.e. a positive determinant means the
row vectors form a right handed system and a negative
determinant means they form a left handed system.
Cross Product
Given two 3D vectors u = <a, b, c>, v = <d, e, f>,
their cross products are defined below
i j k i j k
u x v = det a b c and v x u = det d e f
d e f a b c
The cross product is only defined for two 3D vectors,
the product yield another 3D vector.
62. Determinant and Cross Product
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
63. Determinant and Cross Product
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
i j k
u x v = det 1 2 –1
2 –1 3
64. Determinant and Cross Product
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
i j k i j k i j
u x v = det 1 2 –1 1 2 –1 1 2
2 –1 3 2 –1 3 2 –1
65. Determinant and Cross Product
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
6i –2j –k 4k i 3j
i j k i j k i j
u x v = det 1 2 –1 1 2 –1 1 2
2 –1 3 2 –1 3 2 –1
66. Determinant and Cross Product
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
6i –2j –k 4k i 3j
i j k i j k i j
u x v = det 1 2 –1 1 2 –1 1 2
2 –1 3 2 –1 3 2 –1
= 6i – 2j – k – (4k + i + 3j) or that u x v = 5i – 5j – 5k.
67. Determinant and Cross Product
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
6i –2j –k 4k i 3j
i j k i j k i j
u x v = det 1 2 –1 1 2 –1 1 2
2 –1 3 2 –1 3 2 –1
= 6i – 2j – k – (4k + i + 3j) or that u x v = 5i – 5j – 5k.
* u x v is a orthogonal to both
u and v in the direction determined
by the right-hand rule.
68. Determinant and Cross Product
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
6i –2j –k 4k i 3j
i j k i j k i j
u x v = det 1 2 –1 1 2 –1 1 2
2 –1 3 2 –1 3 2 –1
= 6i – 2j – k – (4k + i + 3j) or that u x v = 5i – 5j – 5k.
* u x v is a orthogonal to both v
u and v in the direction determined
u
by the right-hand rule.
69. Determinant and Cross Product
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
6i –2j –k 4k i 3j
i j k i j k i j
u x v = det 1 2 –1 1 2 –1 1 2
2 –1 3 2 –1 3 2 –1
= 6i – 2j – k – (4k + i + 3j) or that u x v = 5i – 5j – 5k.
* u x v is a orthogonal to both uxv v
u and v in the direction determined
u
by the right-hand rule.
70. Determinant and Cross Product
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
6i –2j –k 4k i 3j
i j k i j k i j
u x v = det 1 2 –1 1 2 –1 1 2
2 –1 3 2 –1 3 2 –1
= 6i – 2j – k – (4k + i + 3j) or that u x v = 5i – 5j – 5k.
* u x v is a orthogonal to both uxv v
u and v in the direction determined
u
by the right-hand rule.
* The length |u x v| is equal to
the area of the parallelogram
defined by u and v.
71. Determinant and Cross Product
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
6i –2j –k 4k i 3j
i j k i j k i j
u x v = det 1 2 –1 1 2 –1 1 2
2 –1 3 2 –1 3 2 –1
= 6i – 2j – k – (4k + i + 3j) or that u x v = 5i – 5j – 5k.
* u x v is a orthogonal to both uxv v
u and v in the direction determined
u
by the right-hand rule.
* The length |u x v| is equal to u x v v
the area of the parallelogram
u
defined by u and v.
72. Determinant and Cross Product
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
6i –2j –k 4k i 3j
i j k i j k i j
u x v = det 1 2 –1 1 2 –1 1 2
2 –1 3 2 –1 3 2 –1
= 6i – 2j – k – (4k + i + 3j) or that u x v = 5i – 5j – 5k.
* u x v is a orthogonal to both uxv v
u and v in the direction determined
u
by the right-hand rule.
* The length |u x v| is equal to u x v v
the area of the parallelogram
u
defined by u and v.
|u x v| = area of the
74. Determinant and Cross Product
Cross Products of i, j, and k
Since the unit coordinate vectors i, j, and k are
mutually perpendicular, we have the following
relations.
75. Determinant and Cross Product
Cross Products of i, j, and k
Since the unit coordinate vectors i, j, and k are
mutually perpendicular, we have the following
relations.
i
+
k j
ixj=k
jxk=i
kxi=j
(Forward ijk)
76. Determinant and Cross Product
Cross Products of i, j, and k
Since the unit coordinate vectors i, j, and k are
mutually perpendicular, we have the following
relations.
i i
+ –
k j k j
ixj=k i x k = –j
jxk=i k x j = –i
kxi=j j x i = –k
(Forward ijk) (Backward kji)
78. Determinant and Cross Product
Algebra of Cross Product
Let u, v and w be 3D vectors:
1. u x u = u x 0 = 0
79. Determinant and Cross Product
Algebra of Cross Product
Let u, v and w be 3D vectors:
1. u x u = u x 0 = 0
2. (Orthogonal Property)
u • (v x u) = 0, v • (v x u) = 0
80. Determinant and Cross Product
Algebra of Cross Product
Let u, v and w be 3D vectors:
1. u x u = u x 0 = 0
2. (Orthogonal Property)
u • (v x u) = 0, v • (v x u) = 0
3. (Anti-commutative) u x v = – (v x u)
81. Determinant and Cross Product
Algebra of Cross Product
Let u, v and w be 3D vectors:
1. u x u = u x 0 = 0
2. (Orthogonal Property)
u • (v x u) = 0, v • (v x u) = 0
3. (Anti-commutative) u x v = – (v x u)
4. If k is a scalar, (ku) x v = u x (kv) = k(u x v)
82. Determinant and Cross Product
Algebra of Cross Product
Let u, v and w be 3D vectors:
1. u x u = u x 0 = 0
2. (Orthogonal Property)
u • (v x u) = 0, v • (v x u) = 0
3. (Anti-commutative) u x v = – (v x u)
4. If k is a scalar, (ku) x v = u x (kv) = k(u x v)
5. (Distributive Law)
(u + v) x w = u x w + v x w
u x (v + w) = u x v + u x w
83. Determinant and Cross Product
Algebra of Cross Product
Let u, v and w be 3D vectors:
1. u x u = u x 0 = 0
2. (Orthogonal Property)
u • (v x u) = 0, v • (v x u) = 0
3. (Anti-commutative) u x v = – (v x u)
4. If k is a scalar, (ku) x v = u x (kv) = k(u x v)
5. (Distributive Law)
(u + v) x w = u x w + v x w
u x (v + w) = u x v + u x w
u
6. (Triple Scalar Product) u•(v x w) = det v
w
84. Determinant and Cross Product
Algebra of Cross Product
Let u, v and w be 3D vectors:
1. u x u = u x 0 = 0
2. (Orthogonal Property)
u • (v x u) = 0, v • (v x u) = 0
3. (Anti-commutative) u x v = – (v x u)
4. If k is a scalar, (ku) x v = u x (kv) = k(u x v)
5. (Distributive Law)
(u + v) x w = u x w + v x w
u x (v + w) = u x v + u x w
u
6. (Triple Scalar Product) u•(v x w) = det v
w
In particular |u•(v x w)| = volume of the parallelepiped
defined by u, v, and w.
85. Determinant and Cross Product
Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w)
86. Determinant and Cross Product
Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w) = u x (2u) + u x (–3w)
87. Determinant and Cross Product
Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w) = u x (2u) + u x (–3w)
= 2(u x u) – 3(u x w)
88. Determinant and Cross Product
Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w) = u x (2u) + u x (–3w)
= 2(u x u) – 3(u x w)
= 2*0 – 3<0, –8, 0> = <0, 24, 0>
89. Determinant and Cross Product
Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w) = u x (2u) + u x (–3w)
= 2(u x u) – 3(u x w)
= 2*0 – 3<0, –8, 0> = <0, 24, 0>
b. u • (2v x 3w) =
90. Determinant and Cross Product
Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w) = u x (2u) + u x (–3w)
= 2(u x u) – 3(u x w)
= 2*0 – 3<0, –8, 0> = <0, 24, 0>
b. u • (2v x 3w) = u • [6(v x w)]
91. Determinant and Cross Product
Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w) = u x (2u) + u x (–3w)
= 2(u x u) – 3(u x w)
= 2*0 – 3<0, –8, 0> = <0, 24, 0>
b. u • (2v x 3w) = u • [6(v x w)]
= 6[u • (v x w)]
92. Determinant and Cross Product
Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w) = u x (2u) + u x (–3w)
= 2(u x u) – 3(u x w)
= 2*0 – 3<0, –8, 0> = <0, 24, 0>
b. u • (2v x 3w) = u • [6(v x w)]
= 6[u • (v x w)]
u
= 6 *det v
w
93. Determinant and Cross Product
Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w) = u x (2u) + u x (–3w)
= 2(u x u) – 3(u x w)
= 2*0 – 3<0, –8, 0> = <0, 24, 0>
b. u • (2v x 3w) = u • [6(v x w)]
= 6[u • (v x w)]
u
= 6 *det v = 6 * 24 = 144
w
Volume of the box
formed by u,v and w
94. Determinant and Cross Product
I don’t have the textbook with me so please do:
HW. From the textbook do all odd on the topic of 2×2
and 3×3 determinants and cross product.