In Physics, we classify quantities into vectors and scalars.
The quantities which have both magnitude and direction are called vectors. Examples are velocity, force, displacement, weight, acceleration, etc.
The quantities which have only magnitude and no direction are called scalar quantities. Examples are mass, volume, speed, time, frequency, etc.
A quantity is said to be a vector if it satisfies the following conditions:
(a) Obeys the law of parallelogram addition.
(b) It has a specified direction.
(c) The addition is commutative, i.e., A+B = B+A
Table of Contents
Representation of a Vector
A vector is represented by a line with an arrowhead. The point O from which the arrow starts is called the tail or initial point, or origin of the vector. Point A, where the arrow ends, is called the tip or head or terminal point of the vector. A vector displaced parallel to itself remains unchanged. If a vector is rotated through an angle other than 3600, it changes.
A vector can be replaced by another when its direction and magnitude are the same.
Unit Vector
A vector having a unit magnitude is called a unit vector. It is used to denote the direction of a given vector.
\(\begin{array}{l}\vec{A}=\hat{a}A\end{array} \)
\(\begin{array}{l}\hat{a}\ \text{is the unit vector along the direction of}\ \vec{A}\end{array} \).
Types of Vectors
(i) Negative of a vector: It has the same magnitude but opposite direction of the given vector.
(ii) Equal vectors: If two vectors have equal magnitude and direction, they are equal vectors.
(iii) Collinear vectors: Two vectors acting along the same straight lines or along parallel straight lines in the same direction or in the opposite direction are called collinear vectors.
(iv) Coplanar vectors: If three or more vectors lie in the same plane, then they are called coplanar vectors.
(v) Zero vector: It is a vector with zero magnitude and no specific direction.
Addition of Vectors
Law of triangle: If two sides of a triangle are shown by two continuous vectors (vector A and vector B), then the third side of the triangle in the opposite direction shows the resultant of two vectors (vector C).
\(\begin{array}{l}\vec{C} = \vec{A}+\vec{B}\end{array} \)
Vector addition is commutative.
⇒ A + B = B + A
Vector addition is associative.
⇒ A+(B+C) = (A+B)+C
If all sides of a polygon are represented by continuous vectors, the vector sum of all sides is zero.
Polygon method
We use this method when we have to add more than two vectors. It is an extension of the triangular law of vector addition. If a number of vectors can be represented in magnitude and direction by the sides of a polygon taken in the same order, then their resultant is represented in magnitude and direction by the closing side of the polygon taken in the opposite order.
Subtraction of vectors
While doing the subtraction of vectors, we change the direction of the vector to be subtracted and then add.
\(\begin{array}{l}\vec{A}-\vec{B}=\vec{A}+(\overrightarrow{-B})\end{array} \)
Null vector
If vector A is multiplied by zero, we get a vector whose magnitude is zero, called a null vector or zero vector. The unit of the vector does not change on being multiplied by a dimensionless scalar.
Properties of null vector
1) It has an arbitrary direction.
2) It is represented by a point.
3) It has zero magnitude.
4) Dot product of a null vector with any vector is always zero.
5) Cross product of a null vector with any vector is also a null vector.
6) When a null vector is added or subtracted from a given vector, the resultant vector is the same as the given vector.
Orthogonal unit vectors
The unit vectors along the X-axis, Y-axis, and Z-axis of the right-handed cartesian coordinate system are written as
\(\begin{array}{l}\hat{i},\hat{j}\ \text{and}\ \hat{k}\end{array} \)
, respectively. They are known as orthogonal unit vectors.
Components of a vector
Consider a vector, V. The components of a vector in a 2D coordinate system are considered to be x-component and y-component. We can represent V = (vx, vy). Let θ be the angle formed between the vector V and the x-component of the vector. The vector V and its x-component (vx) form a right-angled triangle if we draw a line parallel to the y-component (vy).
The horizontal component vx = V cos θ.
The vertical component vy = V sin θ.
Position Vectors
A position vector is a vector that gives the position of a point with respect to the origin of the coordinate system. The magnitude of the position vector is the distance of the point P from the origin O. Vector OP is the position vector that gives the position of the particle with reference to O.
Consider point P, whose coordinates are (x, y).
\(\begin{array}{l}\overrightarrow{OP} = \vec{r}\end{array} \)
x = r cos θ, y = r sin θ
\(\begin{array}{l}\vec{r} = x\hat{i}+y\hat{j}\\\end{array} \)
\(\begin{array}{l}\left | \vec{r}\right |= \sqrt{x^{2}+y^{2}}\end{array} \)
For a point P(x, y, z) in a 3D coordinate system,
\(\begin{array}{l}\overrightarrow{OP}\ \text{or vector r is the position vector with origin O as the initial point.}\end{array} \)
Magnitude of vector OP is:
\(\begin{array}{l}\left | \overrightarrow{OP}\right |=\sqrt{x^{2}+y^{2}+z^{2}}\end{array} \).
Parallelogram Law of Vectors
If two vectors act along two adjacent sides of a parallelogram having a magnitude equal to the length of the sides, both pointing away from the common vertex, then the resultant is given by the diagonal of the parallelogram passing through the same common vertex and in the same sense as the two vectors.
\(\begin{array}{l}\vec{P}+\vec{Q}=\vec{R}\end{array} \)
Or
\(\begin{array}{l}\vec{OA}+\vec{OB}=\vec{OC}\end{array} \)
Dot Product of Two Vectors
If vector A and vector B are two given vectors and θ is the angle between them, then
\(\begin{array}{l}\vec{A}.\vec{B}=AB\cos \theta\end{array} \)
A and B are the magnitudes of vector A and vector B.
AB cos θ is a scalar quantity. B cos θ is the component of vector B in the direction of vector A.
The dot product of two vectors is the product of the magnitude of one vector with the resolved component of the other in the direction of the first vector.
The dot product is also called a scalar product.
Properties of Dot Product
1) Dot product of two vectors is commutative,
i.e.,
\(\begin{array}{l}\vec{A}.\vec{B} = \vec{B}.\vec{A} =AB\cos \theta\end{array} \)
2) The dot product of a vector to itself is the magnitude squared of the vector.
i.e.,
\(\begin{array}{l}\vec{A}.\vec{A}=AA\cos 0= A^{2}\end{array} \)
3) The dot product of two mutually perpendicular vectors is zero.
\(\begin{array}{l}\vec{A}.\vec{B}=AB\cos 90=0\\\end{array} \)
\(\begin{array}{l}\hat{i}.\hat{i}=1\\\end{array} \)
\(\begin{array}{l}\hat{j}.\hat{j}=1\\\end{array} \)
\(\begin{array}{l}\hat{k}.\hat{k}=1\end{array} \)
4) Dot product is distributive.
\(\begin{array}{l}\vec{A}.(\vec{B}+\vec{C})=\vec{A}.\vec{B}+\vec{A}.\vec{C}\end{array} \)
5) The scalar product of two parallel vectors is equal to the product of their magnitudes.
\(\begin{array}{l}\vec{A}.\vec{B}=AB\cos 0= AB\end{array} \)
Cross Product of Two Vectors
The cross product of two vectors, A and B, is denoted by A × B. Its resultant vector is perpendicular to A and B. Cross products are also called vector products. The cross product of two vectors will give the resultant a vector and can be calculated using the right-hand rule.
\(\begin{array}{l}\vec{A}\times \vec{B}=AB\sin \theta \hat{n}\end{array} \)
Consider two vectors, A and B. The cross product of A and B is a vector having a magnitude equal to the product of the magnitudes of the two vectors and the sine of the angle between them and having the direction perpendicular to the plane containing these vectors. θ is the angle between them, and
\(\begin{array}{l}\hat{n}\end{array} \)
is the unit vector perpendicular to the plane of vector A and vector B.
Properties of Cross Product
(a) Cross product is not commutative. Consider two vectors, A and B.
\(\begin{array}{l}\vec{A}\times \vec{B}\ne \vec{B}\times \vec{A}\end{array} \)
\(\begin{array}{l}\vec{A}\times \vec{B}= -\vec{B}\times \vec{A}\end{array} \)
(b) Cross product is distributive. Consider three vectors, A, B, and C.
A × (B + C) = A × B + A × C
(c) The cross product of two parallel vectors is a zero vector.
\(\begin{array}{l}\vec{A}\times \vec{B}=AB\sin \theta \hat{n} = 0\end{array} \)
For parallel vectors, θ = 0. So, sin θ = 0.
Thus, A × B = 0
(d) The cross product of a vector by itself is a null vector.
A × A = 0
(e) The magnitude of the cross product of 2 vectors that are at right angles is equal to the product of the vectors.
If θ = 900,
\(\begin{array}{l}\vec{A}\times \vec{B}=AB\sin 90 \hat{n} = AB\: \hat{n}\end{array} \)
(f) The cross product of unit orthogonal vectors, i, j, k, is as follows:
(i)
\(\begin{array}{l}\hat{i}\times \hat{j}=\hat{k}\\\end{array} \)
\(\begin{array}{l}\hat{j}\times \hat{k}=\hat{i}\\\end{array} \)
\(\begin{array}{l}\hat{k}\times \hat{i}=\hat{j}\\\end{array} \)
\(\begin{array}{l}\hat{j}\times \hat{i}=-\hat{k}\\\end{array} \)
\(\begin{array}{l}\hat{k}\times \hat{j}=-\hat{i}\\\end{array} \)
\(\begin{array}{l}\hat{i}\times \hat{k}=-\hat{j}\end{array} \)
(ii)
\(\begin{array}{l}\hat{i}\times \hat{i}=0\\\end{array} \)
\(\begin{array}{l}\hat{j}\times \hat{j}=0\\\end{array} \)
\(\begin{array}{l}\hat{k}\times \hat{k}=0\end{array} \)
(g) The vector product can be expressed in terms of determinants.
Let
\(\begin{array}{l}\overrightarrow{A}=A_{x}\hat{i}+A_{y}\hat{j}+A_{z}\hat{k}\\\end{array} \)
\(\begin{array}{l}\overrightarrow{B}=B_{x}\hat{i}+B_{y}\hat{j}+B_{z}\hat{k}\\\end{array} \)
\(\begin{array}{l}\overrightarrow{A}\times \overrightarrow{B}=\begin{vmatrix} \hat{i} & \hat{j} &\hat{k} \\ A_{x}& A_{y}& A_{z}\\ B_{x} & B_{y}& B_{z} \end{vmatrix}\end{array} \)
Projection of a Vector
\(\begin{array}{l}\text{The projection of a vector}\ \vec{A}\ \text{on other vector}\ \vec{B}\ \text{is}\ \frac{\vec{A}.\vec{B}}{\left | \vec{B} \right |}.\end{array} \)
\(\begin{array}{l}\text{If}\ \hat{i}\ \text{is a unit vector along a line L, the projection of vector a on the line L is given by}\ \vec{a}.\hat{i}\end{array} \)
\(\begin{array}{l}\text{The projection vector of}\ \vec{AB}\ \text{is zero, if}\ \theta = \frac{\pi}{2}.\end{array} \)
\(\begin{array}{l}\text{The projection vector of}\ \vec{AB}\ \text{is}\ \overrightarrow{BA},\ \text{if}\ \theta = \pi.\end{array} \)
Points to remember
(1) The sum of three non-coplanar forces cannot be zero.
(2) The minimum number of equal forces required for a zero resultant is two.
(3) The minimum number of unequal forces required for a zero resultant is three.
(4) Multiplication of velocity vector by time gives the displacement.
(5) The unit of a vector is not changed if it is multiplied by a dimensionless scalar.
Solved Examples
Question 1. Given A = 2i + 3j and B = i + j. The component of vector A along vector B is
(a) 1/√2
(b) 3/√2
(c) 5/√2
(d) 7/√2
Solution:
The component of vector A along B is A.B/|B|
= (2i + 3j).(i + j)/√(1 + 1)
= 5/√2
Hence, option c is the answer.
Question 2. A force is inclined at an angle of 60° to the horizontal. If its rectangular component in the horizontal direction is 50 N, then the magnitude of the force in the vertical direction is
(a) 25 N
(b) 75 N
(c) 87 N
(d) 100 N
Solution:
Given the horizontal component of force, fx = 50 N
Given angle = 600.
tan 60 = fx/fy
√3 = 50/fy
So fy = 50√3
= 87 N
Hence, option c is the answer.
Question 3. How many minimum number of coplanar vectors having different magnitudes can be added to give zero resultant?
(a) 2
(b) 3
(c) 4
(d) 5
Solution:
According to the Triangle Law of vector addition, a minimum of three vectors is needed to get zero resultant. A minimum of 3 coplanar vectors is required to represent the same physical quantity with different magnitudes that can be added to give zero results.
Hence, option b is the answer.
Question 4. The square of the resultant of two equal forces is three times their product. The angle between the forces is
(a) π
(b) π/3
(c) π/4
(d) π/2
Solution:
Let A and B be the two forces and θ be the angle between them.
Also, A = B.
Given the square of the resultant of two equal forces is three times their product.
F2res = 3AB
⇒ F2res = A2 + B2 + 2AB cos θ
⇒ 3AB = A2 + B2 + 2AB cos θ
Since A = B,
⇒ 3A2 = 2A2 + 2A2cos θ
⇒ A2 = 2A2cos θ
⇒ cos θ = 1/2
⇒ θ = π/3
Hence, option b is the answer.
Question 5. If A = B + C and the values of A, B, and C are 13, 12, and 5, respectively, then the angle between A and C will be
(a) cos-1(5/13)
(b) cos-1(13/12)
(c) π/2
(d) sin-1(5/12)
Solution:
Given A = B + C
Here, 132 = 122 + 52.
So, according to the Pythagoras theorem, the angle between B and C is 90°.
The angle between A and C = cos θ = 5/13.
⇒ θ = cos-1(5/13)
Hence, option a is the answer.
Question 6: If the scalar and vector products of two vectors, A and B, are equal in magnitude, then the angle between the two vectors is
(a) 45°
(b) 90°
(c) 180°
(d) 360°
Solution:
Given A.B = A × B
⇒ AB cos θ = AB sin θ
⇒ tan θ = 1
⇒ θ = 45°
Hence, option a is the answer.
Question 7: If the angle between vectors A and B is θ, then the value of the product (B × A).A is equal to
(a) BA2 cos θ
(b) BA2 sin θ
(c) zero
(d) BA2 sin θ cos θ
Solution:
B × A will be perpendicular to both A and B.
(B × A).A = (B × A)A cos θ (here θ = 90°)
= |B × A||A| cos 90°
= 0
Hence, option c is the answer.
Practice Problems
1. If the angle between two vectors A and B is 1200, then its resultant C will be
(a) C = |A – B|
(b) C < |A – B|
(c) C > |A – B|
(d) C = |A + B|
2. Three concurrent coplanar forces, 1 N, 2 N, and 3 N, acting along different directions on a body
(a) can keep the body in equilibrium if 2 N and 3 N act at right angles
(b) can keep the body in equilibrium if 1 N and 2 N act at right angles
(c) cannot keep the body in equilibrium
(d) can keep the body in equilibrium if 1 N and 32 N act at right angles
3. Vector A has a magnitude of 5 units and lies in the XY-plane, and points in a direction 1200 from the direction of increasing X. Vector B has a magnitude of 9 units and points along the Z-axis. The magnitude of cross product A x B is
(a) 30
(b) 35
(c) 40
(d) 45
4. If a.b = |a×b|, then θ will be
(a) 45°
(b) 60°
(c) 30°
(d) 75°
5. The direction of A is vertically upward, and the direction of B is in the North direction. The direction of A × B will be
(a) western direction
(b) eastern direction
(c) vertically downward
(d) at 450 upward north direction
6. A motorboat covers a given distance in 6 h moving downstream on a river. It covers the same distance in 10 h moving upstream. The time it takes to cover the same distance in still water is
(a) 9 h
(b) 7.5 h
(c) 6.5 h
(d) 8 h
Frequently Asked Questions
Q1
State the Polygon law of vector addition.
The Polygon law of vector addition states that if a number of vectors can be represented in magnitude and direction by the sides of a polygon taken in the same order, then their resultant is represented in magnitude and direction by the closing side of the polygon taken in the opposite order.
Q2
Define vectors and scalars. Give two examples of each.
Vectors are quantities that have both magnitude and direction. Examples are displacement, weight, acceleration, etc. Scalars are quantities that have only magnitude and no direction. Examples are speed, time, frequency, etc.
Q3
What are equal vectors?
Vectors that have the same magnitude and the same direction are called equal vectors. These vectors may start at different positions.
Q4
State the parallelogram law of vector addition.
The parallelogram law of vector addition states that if two vectors act along two adjacent sides of a parallelogram having magnitudes equal to the length of the sides, both pointing away from the common vertex, then the resultant is represented by the diagonal of the parallelogram passing through the same common vertex and in the same sense as the two vectors.
Q5
What are the different types of vectors?
The different vectors are unit vector, zero vector, equal vector, collinear vector, co-planar vector, position vector, co-initial vector, and like and unlike vectors.
