2. Scalar Vector
• A quantity is scalar if
• it has magnitude only.
• A quantity is vector if
• it has magnitude and direction.
• 2 parallel lines have
• the same direction.
3. Vectors
• Two vectors are equal if and only if the
quantities they represent have:
• 1) the same magnitude
• 2) the same direction
• A vector with magnitude 0 and no direction
is also possible.
• It is called the zero vector, 0.
• The opposite of AB is –AB or BA
4. Magnitude
• The size of AB, called its magnitude, or
norm, is written ║AB║
• ║AB║= √((x2-x1)2 + (y2-y1)2)
• If v (c,d), then║v║= √((c)2 + (d)2)
• A unit vector has length 1.
• A zero vector has length 0.
5. Exam QuestionGiven the three vectors u, v, and w.
v = (-2, -3)
u and w are represented in the Cartesian plane below:
x
y
u
w
Which of the following statements is TRUE?
A)
v
and -
u
are opposite.
B)
u
and
v
are equivalent.
C)
w
and (
v
+
w
) are perpendicular.
D)
u
and 3
v
are collinear.
6.
7. Relations Betwixt 2 Vectors
• If 2 vectors are perpendicular, they are
called orthogonal.
• Two vectors that are parallel are also
collinear.
• If one vector can be expressed
• in terms of another,
• (4,6) = 2(2,3), e = 2f,
• they are linearly dependent.
8. Exam Question
Quadrilateral RSTU is a parallelogram and M is
the point of intersection of its diagonals.
Antoine lists the following vector operation
statements:
S
R
T
U
M
1)
MU2SRST
2)
SM2URUT
3)
RTRURS
4)
0MUMSMRMT
5)
RTSTSR
Which of these statements are true?
A) 1, 2 and 3 only C) 2, 4 and 5 only
B) 1, 2 and 5 only D) 1, 3 and 4 only
9. Exam Question
Givena andb , two vectors illustrated below.
b
a
Which one of the following diagrams illustrates the relation betweena andb andr , the
resultant vector?
A)
a
b
r
C)
a
b
r
B)
a
b
r
D)
a
r
b
11. Operations on Vectors
• Vectors can moved around.
• They must be added tip to tail.
• The resultant of u+v = v+u
• If u = (a,b), and v = (c,d), then
• u + v = (a+c, b+d)
• u- v= (a-c, b-d)
• The multiplication by k of a vector, v
• kv = (kc, kd), ║kv║= ║k║v║
12. Scalar (Dot) Product
• The scalar product of two vectors,
• uv where u = (a,b), v = (c,d)
• = ac + bd = ║u║v║cosΘ,
• Where Θ is the angle between the 2 vector
tails.
• When Θ = 90˚, cos Θ = 0, so the scalar
product is 0.
• Therefore the scalar product of two
perpendicular vectors is 0.
13. Multiplication of a Vector
• The product of a vector and a scalar is
always a vector.
• Associative: a(bv) = (ab)v
• The existence of the scalar, k=1,
is used as an identity element, so that
1v= v
• Distributivity: a(u+v) = au+av
• Distributivity: (a+b)v = av+bv
14. Exam Question
Given the following information:
a andb are nonzero vectors in the plane
ba
k is a scalar not equal to zero
k 1
Which of the following statements is true?
A) bkak)bak(
B) bkak)bak(
C) If
0ba
thena andb are collinear
D) Ifa = kb thena andb are noncollinear
15. Basis Vectors
• Any vector can be broken down into a sum of 2
other vectors, which in turn can be broken down
into a product of a scalar and a vector.
• A linear combination defines a vector by using
other vectors defined previously.
• The real numbers used are the coefficients of
the combination.
• Vector (3,4) = (3,0) + (0,4)
• =3(1,0) + 4(0,1)
• = 3 i + 4 j
• Where i = (1,0) and j = (0,1), coefficients 3,4
16. Exam Question
The Egyptians used an ingenious pulley system to move the blocks of stone used in the
construction of pyramids. To minimize the work needed to displace the blocks, they
applied a force oriented at 26. (Work (Nm) is the scalar product of the force vector and
the displacement vector.)
26
200 m
1500 N
Rounded to the nearest Nm, what work is needed to displace a block of stone horizontally
for a distance of 200 m, if the force applied to it is 1500 N oriented at 26o
?
A) 131 511 Nm C) 228 768 Nm
B) 194 076 Nm D) 269 638 Nm
18. Exam Question
Given )4-,1(vand),2,3(u
What are the components of the resultant of the following vector operation?
v2u
A) (1, 10) C) (2, 6)
B) (1, -6) D) (5, -6)
19. Exam Question
Given the following prism having a rectangular
base.
A B
D C
E F
H
G
Which vector is equivalent to the resultant of the expressionAD +HE +AE ?
A) DH C) FB
B) BE D) BC
21. Summary
• A vector is a quantity that has a size and a
direction and is represented by an arrow
representing its characteristics.
• AB corresponds with –BA
• The size of AB, called its magnitude, or
norm, is written ║AB║
• ║AB║= √((x2-x1)2 + (y2-y1)2)
• If v (c,d),║v║= √((c)2 + (d)2)
22. Summary
• Zero vector is a vector with magnitude 0.
• It is written 0
• Two vectors can be equal, opposite,
collinear (in-line) or non-collinear, or
orthogonal (perpendicular).
• A resultant is the result of combining two
or more vectors, tip to tail.
• Subtracting a vector equals adding its
opposite.
23. Summary
• Chasles’ Rule, AB + BC = AC
• If u = (a,b), and v = (c,d), then
• u + v = (a+c, b+d)
• u- v= (a-c, b-d)
• The multiplication by k of a vector, v
• kv = (kc, kd), ║kv║= ║k║v║
• The scalar product of two vectors uv = ac
+ bd = ║u║v║cosΘ,
• Where Θ is the angle between the 2 vector tails.
24. Summary
• The basis of a the Cartesian plane is x
and y. This can also be expressed as 2
vectors i (1,0) and j (0,1). These
vectors are orthogonal or perpendicular to
each other.
• Other vectors may for the basis for other
vectors.
