Unlocking the Potential: Deep dive into ocean of Ceramic Magnets.pptx
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Vector and scalar
1. Introduction to scalar and vector
Let us begin with the following questions:
a. You are at rest position and an object is moving at a speed of 10km/hr. What would be the
distance (gap) between you and the object after 30 minutes?
It is pretty simple. The distance will be 5 km.
b. You are moving to the opposite of an object at a speed of 20 km/hr. The object is moving at
a speed of 10km/hr in the North.
(i) What would be the distance covered by you in 30 minutes?
(ii) What would be the distance (gap) between you and the object in 30 minutes?
(iii) What would be the displacement of the object in 30 minutes?
(iv) What would be the displacement of the object w.r.t the origin in 15 minutes?
(i) It is 10 km.
(ii) In 30 minutes, you will cover a distance of 10 km (in the South) and the object will cover a
distance of 5 km (in the North). So, a gap of 15 km will be made.
(iii) Actually, the question is not precise. It should have been asked in this fashion, 'What
would be the displacement of the object w.r.t you after 30 minutes'? The displacement w.r.t
you will be 15 km in the case.
(iv) It is 2.5 km to the North of the origin. (Note: displacement is a vector & direction must be
specified for vectors)
Q. How are magnitude and direction imparted to a vector quantity?
Magnitude and direction are imparted on the Cartesian plane. To impart the magnitude, a
reference point or an initial point is required as the magnitude is the distance between two
points. While, to impart the direction, a line is required that passes through the reference
point. It will become clear from the following diagram:
2. The magnitude of π
β = |π|
ββββββ = OA. OA can be evaluated if the coordinates of O and A are Known.
If O = (0,0) and A = (a,b), OA = βπ2 + π2. It is not mandatory that the direction (ΞΈ) must be
always about the horizontal line or x-axis. It can be about any line. But the line must be
specified. For example, a ball is moving at a velocity of 5km/hr in the North or in the N45E.
Here the line is specified. N45E is 45Β° about the North in clockwise direction. Here the line is
specified as the North.
So now on, we must be able to decide that the vector π
β has the magnitude |π|
ββββββ and it is
directed somewhere.
Q. Look at the diagram, there are two vectors equal in magnitude. Are
the vectors equal?
No. Not only magnitude but also the direction of two vectors must be
same in order tobe equal.
Representation of vectors & resolution of vectors
Vectors can be represented in the coordinate form (p,q) and in the form of unit vectors. If π
β
and π
β are unit vectors along x-axis and y-axis respectively,
π
β = pπ
β & π
β = qπ
β
ΞΈ
O
A
X
Y
π
β
o
3. This can be written only if π
β and π
β have same direction and if π
β and π
β have same direction.
Here, π
β and π
β are called horizontal and vertical components of the vector π
β respectively. The
phenomenon of breaking down the vector into 2 components (in 2D) is called vector
resolution.
In the coordinate form vector π
β is represented as π
ββββ = (p, q).
Note: Direction also must be specified in both forms.
Unit vectors have magnitude 1. In the above figure, π
β & π
β are assumed to be greater than their
corresponding unit vectors: π
β = pπ
β & π
β = qπ
β
Using triangle law of vector addition for above example,
Here, the magnitude of π
β = |π|
ββββββ = |π
β + π
β| = |ππ
β + ππ
β| = distance of OA
π
β
(p,q)
ΞΈ
O
A
X
Y
π
β
π
β
π
β
π
β
π
β
π
β =pπ
β
π
β =qπ
β
π
β = π
β + π
β = pπ
β+ qπ
β
O
A
4. From above, we can infer that a vector (p, q) = ππ
β + ππ
β has:
a horizontal component = π
ββββ = ππ
β and a vertical component = π
ββββ = ππ
β
Or, we can say that a vector is equal to the summation of it's components.
Q. Can the resultant of two different vector quantities (example force and velocity) be
evaluated?
No.
We cannot decide whether the resultant (π
ββ) would be force (πΉ
β) or velocity ( π
ββ) or any other
quantity.
Q. Give a justification for the statement 'Both the horizontal and vertical components of a
vector are vectors'.
Directions of the components of a vector can be identified w.r.t the vector itself or w.r.t the x-
axis or w.r.t the y-axis.
Q. How would a vector origin?
A vector gets originated due to:
οΆ The resultant of 2 vectors
οΆ A single vector
οΆ Cross/vector product between 2 vectors
Let us have a look on the diagram that shows the distribution of forces at 4 different points
outside the bounded region. An object of mass 0.5 kg is inside the bounded region and
exerting equal force (5N) opposite to the forces as shown below:
πΉ
β
π
ββ
π
ββ
5. At all the points, the mass of the object will remain unchanged as mass being scalar quantity is
not affected by direction. But the resultant force at all the points will remain changed.
Resultant of vectors
1. Imagine a ball is being pulled with two forces in opposite directions. Where would the ball
go?
The ball will move to the right side since the net force or the resultant of two forces is directed
along the right side.
The net/resultant force = 75-10 = 65N and the direction is along the direction of P.
2N
4N
6N
8N
5N
5N
5N
5N
π
ββ = 10N π
ββ = 75N
6. 2. Imagine a ball is being pulled with two forces in same direction. Where would the ball go?
The ball will move to the right side since the net force or the resultant of two forces is directed
along the right side.
The net/resultant force = 75+10 = 85N and the direction is along the direction of P.
In the problem 1: the angle between two forces is 180Β° and the resultant force is directed
along the greater force. And, the magnitude of the resultant force = greater force-smaller
force.
In the problem 2: the angle between two forces is 0Β° and the resultant force is directed along
the given forces. And, the magnitude of the resultant force = greater force + smaller force.
3. Imagine a ball is being pulled with two forces in different directions. Where would the ball
go?
To answer this we need to observe the previous problems. From those problems we can infer
that the magnitude and the direction of the resultant force depend on: the angle between two
forces and the magnitude of the forces.
75N
10N
10N
75N
7. 4. What would be the direction of the resultant force? Just think of intuitive answer.
The direction of the resultant force is always closer to the greater force and farther from the
smaller force as the impact of the greater force is greater than that of the smaller force. The
ball neither moves along 10N nor along 75N. Hence the ball moves midway between two
forces being closer to the 75 N.
5. What would be the magnitude of the resultant force (π
ββ) if ΞΈ = 60Β°?
The resultant force (π
ββ) will be such that R2
= |π|
ββββββ2
+|π|
ββββββ2
+2|π|
ββββββ|π|
ββββββCosΞΈ
where: π
ββ and π
ββ are two forces
ΞΈ is the angle between two forces
|π|
ββββββ & |π|
ββββββ are the magnitudes of π
ββ & π
ββ respectively.
In this example, π
ββ = 75π
βββββββββ, π
ββ = 10π
βββββββββ, ΞΈ = 60Β°
R2
= |75|
βββββββββ2
+|10|
βββββββββ2
+2|75|
βββββββββ|10|
βββββββββCos60 = 100+5625+2.10.75.0.5 = 5725+750= 6475 N2
ΞΈ
10N
75N
π
ββ
8. Thus, π
ββ = β6475 N = 80.46 N
Since, the force is a vector quantity it must have direction too. Let Ο be the angle between π
ββ &
75π
βββββββββ. Then Ο will be the direction of π
ββ w.r.t. the line of 75π
βββββββββ. We can assign the direction for π
ββ
w.r.t. 10π
βββββββββ too.
The direction of π
ββ is such that: tan Ο = QSin ΞΈ/(P+QCos ΞΈ)
In this example, tan Ο = 10.Sin60/(75+10.Cos60) or, tan Ο = 8.66/80, or Ο = π‘ππβ1
(8.66/80) =
6Β°.
Meaning, π
ββ & 75π
βββββββββare at a gap of 6Β° while, π
ββ & 10π
βββββββββare at a gap of 54Β°. It shows that the
impact of 75π
βββββββββ is higher on the ball.
6. What would happen if ΞΈ = 80Β° instead of 60Β°in example 5?
The angular gap of two vectors is greater for ΞΈ = 80Β°. Whatever the angular gap is, similar
phenomenon occurs. Meaning, the greater force will have greater impact on the ball and the
smaller force will have smaller impact. ΞΈ = 80Β° will occur if 10 N further rotates towards N in
anticlockwise. During rotation of 10 N (to maintain ΞΈ = 80Β°), the resultant will get pulled a bit
towards 10 N. This will lead in increasing of the value Ο = 6Β°.
7. Let us decrease the value of π
ββ progressively until π
ββ becomes 0 keeping π
ββ and ΞΈ unchanged.
What would happen to π
ββ?
The magnitude and direction of π
ββ also changes progressively. Ultimately, π
ββ becomes π
ββ as
there will be no π
ββ to impart impact on the ball.
9. In the last example, the magnitude of π
ββ = |π |
ββββββ = |π|
ββββββ = 10N and the direction of π
ββ w.r.t 0N is Ο =
60Β° or 0Β° w.r.t. 10N.
We can also evaluate the resultant vector by resolving the vectors in horizontal and vertical
components as shown below:
10N
75N
π
ββ
10N
40N
π
ββ
10N
20N
π
ββ
10N
0N
R =
10N
H
V
10. There are two vectors denoted by blue and red line arrows. Both have horizontal and vertical
components. Just add the horizontal component of one vector to the horizontal component of
another vector. Do similar in case of their vertical components. The two components will be
formed. The angle between these newly formed components will be 90Β°. Just put these
components in the formula of resultant vector.
Note: To study the net impact of 2 vectors on a body, the resultant of two vectors is necessary
to identify. The magnitude and direction of the resultant vector are to be evaluated. The
calculation of the magnitude is different to that of the scalar quantities.
Dot product and vector product
(i) Derivation of dot/scalar product between two vectors in 2D
Let π
β
ββ and π
β
ββ be two vectors with the origin 'o' as the common reference or initial point. Let ΞΈ2
and ΞΈ1 be their corresponding directions as show in the figure. Let ΞΈ2 be greater than ΞΈ1 for
instance. Hence, the angle between the vectors = ΞΈ2 β ΞΈ1 = ΞΈ. Points P and Q have been
projected onto the x-axis normally. Thus, the two right angled triangles, βOPM and βOQN
get developed.
From βOPM and βOQN,
11. CosΞΈ2 = m/OP = m/βππ + ππ = m/|π
β
ββ|β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦..1
CosΞΈ1 = n/OQ = n/βππ + ππ = n/|π
β
ββ|β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦..β¦.2
SinΞΈ2 = p/OP = p/βππ + ππ = p/|π
β
ββ|β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.3
SinΞΈ1 = q/OQ = q/βππ + ππ = q/|π
β
ββ|β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦.β¦4
Taking Cos to both sides of the expression ΞΈ2 β ΞΈ1 = ΞΈ,
Cos ΞΈ = Cos (ΞΈ2 β ΞΈ1) = Cos ΞΈ2 Cos ΞΈ1+ Sin ΞΈ2SinΞΈ1β¦β¦β¦β¦β¦β¦β¦.β¦β¦.5
From 1, 2, 3, 4 & 5, mn+pq = |π
β
ββ||π
β
ββ| CosΞΈβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦..6
Let π
β and π
β be two unit vectors along x-axis and y-axis respectively. Vectors can be
represented in the form of unit vectors as:
π
β
ββ = (m, p) = mπ
β+pπ
β =π
ββββ+π
β
βββ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.7
π
β
ββ = (n, q) = nπ
β+qπ
β
ββ = π
βββ+π
β
βββ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦8
From 7 & 8,
mπ
β = π
ββββ, nπ
β = π
βββ, pπ
β = π
β
ββ and qπ
β
ββ =π
β
ββ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦9
Multiplying 7 and 8,
π
β
ββ .π
β
ββ = (mπ
β+pπ
β) . (nπ
β+qπ
β
ββ)
π
β
ββ .π
β
ββ = mn (π
β .π
β ) + mq (π
β .π
β ) + pn (π
β .π
β ) + pq (π
β .π
β )β¦β¦β¦β¦β¦.β¦β¦β¦..10
For, π
β .π
β = π
β .π
β = π and π
β .π
β = π
β .π
β = 0 equation 10 becomes,
π
β
ββ .π
β
ββ = mn + pqβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ β¦β¦11
From 11 & 6,
π
β
ββ .π
β
ββ = mn+pq = |π
β
ββ||π
β
ββ| Cos ΞΈ iff π
β .π
β = π
β .π
β = π and π
β .π
β = π
β.π
β = 0
(ii) Derivation of cross/vector product between two vectors in 2D
Taking Sin to both sides of the expression ΞΈ2 β ΞΈ1 = ΞΈ,
Sin ΞΈ = Sin (ΞΈ2 β ΞΈ1) = Sin ΞΈ2 Cos ΞΈ1 - CosΞΈ2SinΞΈ1β¦β¦β¦β¦β¦β¦β¦.β¦β¦.12
12. From 1, 2, 3, 4 & 5, pn-mq = |π
β
ββ||π
β
ββ| SinΞΈβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦..13
Let π
β and π
β be two unit vectors along x-axis and y-axis respectively. Vectors can be
represented in the form of unit vectors as:
π
β
ββ = (m, p) = mπ
β+pπ
β =π
ββββ+π
β
βββ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦14
π
β
ββ = (n, q) = nπ
β+qπ
β
ββ = π
βββ+π
β
βββ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦15
From 7 & 8,
mπ
β = π
ββββ, nπ
β = π
βββ, pπ
β = π
β
ββ and qπ
β
ββ =π
β
ββ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦16
Multiplying 7 and 8,
π
β
ββ*π
β
ββ = (mπ
β+pπ
β) * (nπ
β+qπ
β
ββ)
π
β
ββ*π
β
ββ= mn (π
β*π
β) + mq (π
β *π
β ) + pn (π
β * π
β) + pq (π
β*π
β)β¦β¦β¦β¦β¦.β¦β¦β¦.17
For, π
β * π
β = π
β *π
β = π, π
β *π
β = βπ & π
β *π
β = 1 equation 10 becomes,
π
β
ββ*π
β
ββ= pn - mqβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ β¦β¦18
From 11 & 6,
π
β
ββ*π
β
ββ= pn - mq= |π
β
ββ||π
β
ββ| Sin ΞΈ π
Μ iff π
β * π
β = π
β *π
β = π, π
β *π
β = βπ & π
β*π
β = 1
Since, |π
β
ββ||π
β
ββ| Sin ΞΈ is a formula of area and area is a vector whose direction is normal to the
plane, the expression is multiplied by the unit vector π
Μ.