Scalars and Vectors
Scalars quantities: The physical quantities which have only magnitude and no direction are called scalar quantities or scalars.
eg: Mass, volume, density, time, temperature, electric current etc.
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Vector quantities: The physical quantities which have both magnitude and direction and obey the laws of vector addition are called vector quantities or vectors.
eg: Displacement, velocity, force, momentum etc.
Representation of Vectors
A vector quantity is represented by a straight line with an arrowhead over it.
For example, the vector \(A \) is represented by \( \vec{A} \).
Position Vector
A vector which tells the position of an object with reference to the origin of a co-ordinate system is called position vector.
Displacement Vector
It is that vector which tells how much and in which direction an object has changed its position in a given time interval.
\( \scriptsize \overrightarrow{PQ} = \text{Position vector of Q} – \text{Position vector of P} \)
Polar and Axial Vectors
Polar Vectors: The vectors which have a starting point or a point of application are called polar vectors.
eg: Displacement, velocity, force etc.
Axial Vectors: The vectors which represent rotational effect and act along the axis of rotation are called axial vectors.
eg: Angular velocity, torque, angular momentum etc.
Some Definitions in Vector Algebra
- Equal Vectors
Two vectors are said to be equal if they have the same magnitude and the same direction.
\[
\vec{A} = \vec{B}
\]
- Negative of a Vector
The negative of a vector is defined as another vector having the same magnitude but having an opposite direction.
- Modulus of a Vector
The modulus of a vector means the length or the magnitude of that vector.
\[
|\vec{A}| = A
\]
- Unit Vector
A unit vector is a vector of unit magnitude drawn in the direction of a given vector.
\[
\hat{A} = \frac{\vec{A}}{|\vec{A}|}
\]
- Fixed Vector
The vector whose initial point is fixed is called a fixed vector or a localized vector.
- Free Vector
A vector whose initial point is not fixed is called a free vector or a non-localized vector.
eg: Velocity vector of a particle moving along a straight line.
- Collinear Vectors
The vectors which either act along the same line or along parallel lines are called collinear vectors.
Two collinear vectors having the same direction (( \theta = \alpha )) are called like or parallel vectors.
8. Coplanar Vectors
The vectors which act in the same plane are called coplanar vectors.
9. Co-initial Vectors
The vectors which have the same initial point are called co-initial vectors.
10. Co-terminus Vectors
The vectors which have the common terminal point are called co-terminus vectors.
Zero Vector and Its Properties
A zero or null vector is a vector that has zero magnitude and an arbitrary direction. It is represented by \( \vec{0} \).
Properties of Zero Vectors
- When a vector is added to a zero vector, we get the same vector:
\[
\vec{A} + \vec{0} = \vec{A}
\]
- When a real number is multiplied by a zero vector, we get a zero vector:
\[
\lambda \cdot \vec{0} = \vec{0}
\]
- When a vector is multiplied by zero, we get a zero vector:
\[
0 \cdot \vec{A} = \vec{0}
\]
- If \( \lambda \) and \( \mu \) are two different non-zero real numbers, then the relation
\[
\lambda \cdot \vec{A} = \mu \cdot \vec{B}
\]
can hold only if both \( \vec{A} \) and \( \vec{B} \) are zero vectors.
Physical Examples of Zero Vectors
- The position vector of a particle lying at the origin is a zero vector.
- The velocity vector of a stationary object is a zero vector.
- The acceleration vector of an object moving with uniform velocity is a zero vector.
Multiplication of a Vector by a Real Number
When a vector is multiplied by a real number \( \lambda \), we get another vector \( \lambda \vec{A} \). The magnitude of \( \lambda \vec{A} \) is \( \lambda \) times the magnitude of \( \vec{A} \).
Addition or Composition of Vectors
The resultant of 2 or more vectors is that single vector which produces the same effect as the individual vectors together would produce. The process of adding two or more vectors is called composition of vectors.
The following 3 laws of vector addition can be used to add two or more vectors having any inclination to each other:
- Triangle law of vector addition (for adding 2 vectors).
- Parallelogram law of vector addition (for adding 2 vectors).
- Polygon law of vector addition (for adding more than 2 vectors).
1. Triangle Law of Vector Addition
If 2 vectors can be represented by the two sides of a triangle taken in the same order, then their resultant is represented by the third side of the triangle taken in the opposite order.
2. Parallelogram Law of Vector Addition
If 2 vectors can be represented by the 2 adjacent sides of a parallelogram drawn from a common point, then their resultant is represented by the diagonal of the parallelogram drawn from that point.
\[
\vec{R} = \vec{A} + \vec{B}
\]
3. Polygon Law of Vector Addition
If a number of vectors are represented by the sides of an open polygon taken in the same order, then their resultant is represented by the closing side of the polygon taken in the opposite order.
\[
\vec{R} = \vec{A} + \vec{B} + \vec{C} + \vec{D}
\]
Properties of vector addition
- Vector Addition is Commutative:
\[
\vec{A} + \vec{B} = \vec{B} + \vec{A}
\]
- Vector Addition is Associative:
\[
(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})
\]
Flight of a Bird
Flying of a bird as an example of composition of vectors. When a bird flies, it pushes the air with forces \( F_1 \) and \( F_2 \) in the downward direction with its wings \( W_1 \) and \( W_2 \), where the lines of action of these two forces meet at point O.
By Newton’s 3rd law of motion, the air exerts equal and opposite reactions \( R_1 \) and \( R_2 \). According to the parallelogram law, the resultant \( R \) is the resultant of \( R_1 \) and \(R_2 \) on the bird in the upward direction and allows the bird to fly upward.
Working of a Sling
The working of a sling is based on the parallelogram law of vector addition. When a stone held at the point O on the rubber band is pulled, the tensions \( T_1 \) and \( T_2 \) are produced. According to the parallelogram law of forces, the resultant ( R ) is the tension \( T \) and acts on the stone as shown in the figure.
Analytical Treatment of the Parallelogram Law of Vector Addition
Let the two vectors \( \vec{A} \) and \( \vec{B} \) inclined to each other at an angle \( \theta \) be represented both in magnitude and direction by the adjacent sides \( OP \) and \( OQ \) of the parallelogram \( OPQS \). Then according to the parallelogram law, the resultant of \( \vec{A} \) and \( \vec{B} \) is represented by the diagonal \( \vec{R} \) of the parallelogram.
Magnitude of Resultant ( R ):
Draw \(SN \) perpendicular to \(OP \) produced. Then:
\[
\angle SNS’ = \theta, \quad OP = A, \\ \quad PS = B, \quad \vec{R} = \vec{R}.
\]
From right-angled triangle \( \Delta SNP \):
\[
\sin \theta = \frac{SN}{PS} \quad \Rightarrow \quad SN = PS \sin \theta
\]
\[
\cos \theta = \frac{PN}{PS} \quad \Rightarrow \quad PN = PS \cos \theta
\]
Using Pythagoras’ theorem in \( \Delta ONS \):
\[
OS^2 = ON^2 + SN^2
\]
\[
R^2 = (OP + PN)^2 + SN^2
\]
\[
R^2 = (A + B \cos \theta)^2 + (B \sin \theta)^2
\]
Simplifying, we get;
\[
\scriptsize R^2 = A^2 + 2AB \cos \theta + B^2 \cos^2 \theta + B^2 \sin^2 \theta
\]
Since \( \cos^2 \theta + \sin^2 \theta = 1 \), we get:
\[
R^2 = A^2 + B^2 + 2AB \cos \theta
\]
Finally, the magnitude of the resultant vector is:
\[
R = \sqrt{A^2 + B^2 + 2AB \cos \theta}
\]
Special Cases
- If the two vectors \( \vec{A} \) and \( \vec{B} \) are acting along the same direction, \( \theta = 0^\circ \):
\[
\vec{R} = \sqrt{A^2 + B^2 + 2AB \cos \theta}
\]
\[
\cos 0^\circ = 1
\]
\[
\vec{R} = \sqrt{A^2 + B^2 + 2AB} = A + B
\]
The direction of resultant is the same as that of the given vectors.
- If the vectors \( \vec{A} \) and \( \vec{B} \) are acting along mutually opposite directions, \( \theta = 180^\circ \):
\[
\vec{R} = \sqrt{A^2 + B^2 + 2AB \cos 180^\circ}
\]
\[
\cos 180^\circ = -1
\]
\[
\vec{R} = \sqrt{(A – B)^2} = A – B \quad (A > B)
\]
The direction of the resultant vector is in the direction of the bigger vector.
- When the two vectors are acting at right angles to each other, \( \theta = 90^\circ \):
\[
\vec{R} = \sqrt{A^2 + B^2 + 2AB \cos 90^\circ}
\]
\[
\cos 90^\circ = 0
\]
\[
\vec{R} = \sqrt{A^2 + B^2}
\]
Also, the tangent of the angle is:
\[
\tan \alpha = \frac{B \sin 90^\circ}{A + B \cos 90^\circ} = \frac{B}{A}
\]
\[
\alpha = \tan^{-1} \left( \frac{B}{A} \right)
\]
Solved Example: Two forces of ( 5N ) and ( 7N) act on a point with an angle of \( 60^\circ \) between them. Find the resultant force.
\[
\vec{R} = \sqrt{5^2 + 7^2 + 2(5)(7) \cos 60^\circ}
\]
\[
\vec{R} = \sqrt{25 + 49 + 70} = \sqrt{144} = 12 \, \text{N}
\]
Solved Example: Two equal vectors have their resultant equal to either. Show the inclination between them.
\[
\vec{A} = \vec{B} = \vec{R}
\]
\[
\vec{R}^2 = \vec{A}^2 + \vec{B}^2 + 2AB \cos \theta
\]
\[
\vec{R}^2 = 2A^2 + 2A^2 \cos \theta
\]
\[
\vec{R}^2 = 2A^2 (1 + \cos \theta)
\]
\[
\cos \theta = -\frac{1}{2}
\]
\[
\theta = 120^\circ
\]
Resolution of a Vector
It is the process of splitting a vector into two or more vectors in such a way that their combined effect is the same as that of the given vector. The vectors into which the given vector is split are called components of the vector.
Orthogonal Triad of Unit Vectors: Base Vectors
Three unit vectors \( \hat{i}, \hat{j}, \hat{k} \) are used to represent the positive directions of the x-axis, y-axis, and z-axis respectively, and are called unit vectors or base vectors.
These 3 mutually perpendicular unit vectors are collectively known as orthogonal trial of unit vectors or base vectors.
\[
|\hat{i}| = |\hat{j}| = |\hat{k}| = 1
\]
Rectangular Components of a Vector
When a vector is resolved along two mutually perpendicular directions, the components so obtained are called rectangular components of the given vector.
\[
\vec{A} = A_x \hat{i} + A_y \hat{j}
\]
Where \( A_x \) is the horizontal or x-component of \( \vec{A} \), and \( A_y \) is the vertical or y-component of \( \vec{A} \).
\[
A_x = A \cos \theta \quad \text{and} \quad A_y = A \sin \theta
\]
\[
A_x^2 + A_y^2 = A^2 (\cos^2 \theta + \sin^2 \theta)
\]
\[
A = \sqrt{A_x^2 + A_y^2}
\]
\[ \text{and}\ tan\theta = \frac{A_y}{A_x} \]
Note: Rectangular Components of a Vector in 3 Dimensions
\[
\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}
\]
\[
|\vec{A}| = A = \sqrt{A_x^2 + A_y^2 + A_z^2}
\]
Solved Example: Find the vector \( \vec{AB} \) and its magnitude if it has initial point \( (1, 2, -1) \) and final point \( B(3, 2, 2) \).
\[
\text{Position Vector of } A = \hat{i} + 2\hat{j} – \hat{k}
\]
\[
\text{Position Vector of } B = 3\hat{i} + 2\hat{j} + 2\hat{k}
\]
\[
\vec{AB} = \text{P.V. of } B – \text{P.V. of } A
\]
\[
\vec{AB} = (3-1) \hat{i} + (2-2) \hat{j} + (2+1) \hat{k}
\]
\[
\vec{AB} = 2\hat{i} + 3\hat{k}
\]
\[
|\vec{AB}| = \sqrt{2^2 + 3^2} = \sqrt{13}
\]
Solved Example: Find the unit vector of the resultant of the vectors \( \vec{A} = \hat{i} + 4\hat{j} + 3\hat{k} \) and \( \vec{B} = 2\hat{i} – \hat{j} – 5\hat{k} \).
\[
\vec{R} = \vec{A} + \vec{B}
\]
\[
\vec{R} = (1+2) \hat{i} + (4-1) \hat{j} + (3-5) \hat{k}
\]
\[
\vec{R} = 3\hat{i} + 3\hat{j} – 2\hat{k}
\]
The magnitude of \( \vec{R} \):
\[
|\vec{R}| = \sqrt{3^2 + 3^2 + (-2)^2} = \sqrt{18} = 3\sqrt{2}
\]
The unit vector:
\[
\hat{R} = \frac{\vec{R}}{|\vec{R}|} = \frac{3\hat{i} + 3\hat{j} – 2\hat{k}}{3\sqrt{2}}
\]
\[
\implies \hat{R} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} – \frac{2}{3\sqrt{2}} \hat{k}
\]
Scalar Product or Dot Product of Two Vectors
The dot product of two vectors \( \vec{A} \) and \( \vec{B} \) is defined as the product of the magnitudes of \( \vec{A} \) and \( \vec{B} \) and the cosine of the angle \( \theta \) between them:
\[
\vec{A} \cdot \vec{B} = AB \cos \theta
\]
Physical Examples of Scalar Product of 2 Vectors
- Work Done \(W\):
\[
W = \vec{F} \cdot \vec{d}
\]
It is the scalar product of force and displacement vectors.
- Instantaneous Power (P):
\[
P = \vec{F} \cdot \vec{v}
\]
It is the dot product of force and instantaneous velocity.
Properties of Scalar Product
- The scalar product is commutative:
\[
\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}
\]
- The scalar product is distributive over addition:
\[
\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}
\]
- If \( \vec{A} \) and \( \vec{B} \) are two vectors perpendicular to each other, then their scalar product is zero:
\[
\vec{A} \cdot \vec{B} = AB \cos 90^\circ = 0
\]
- The scalar product of a vector with itself is equal to the square of its magnitude:
\[
\vec{A} \cdot \vec{A} = A^2 \cos 0^\circ = A^2
\]
- The scalar product of two similar base vectors is unity, and that of two different base vectors is zero:
\[
\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1
\]
\[
\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0
\]
- If:
\[
\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} \\ \quad \text{and} \quad \vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k}
\]
Then:
\[
\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z
\]
- The angle between two vectors:
\[
\cos \theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|}
\]
Vector Product or Cross Product of Two Vectors
The vector product of two vectors is defined as the vector whose magnitude is equal to the product of the magnitudes of two vectors and sine of the angle between them, and whose direction is perpendicular to the plane of the two vectors and is given by the right-hand rule.
\[
\vec{A} \times \vec{B} = AB \sin \theta \, \hat{n}
\]
where \( \hat{n} \) is a unit vector perpendicular to the plane of \( \vec{A} \) and \( \vec{B} \).
Geometrical Interpretation of Vector Product
\[
|\vec{A} \times \vec{B}| = \text{Area of parallelogram OPQR}
\]
\[
\frac{1}{2} |\vec{A} \times \vec{B}| = \text{Area of triangle OPQ}
\]
Physical Examples of Vector Product
- Torque \( \vec{\tau} \):
It is the vector product of position vector \( \vec{r} \) and force vector \( \vec{F} \).
\[
\vec{\tau} = \vec{r} \times \vec{F}
\]
- Angular Momentum \( \vec{L} \):
It is the cross product of position vector and linear momentum.
\[
\vec{L} = \vec{r} \times \vec{p}
\]
- Instantaneous Velocity \( \vec{v} \):
It is the vector product of angular velocity and the position vector.
\[
\vec{v} = \vec{\omega} \times \vec{r}
\]
Properties of Vector Product
- Vector product is anti-commutative:
\[
\vec{A} \times \vec{B} = – \vec{B} \times \vec{A}
\]
- Vector product is distributive over addition:
\[
\vec{A} \times (\vec{B} + \vec{C}) = (\vec{A} \times \vec{B}) + (\vec{A} \times \vec{C})
\]
- Vector product of two parallel or antiparallel vectors is a null vector:
\[
\vec{A} \times \vec{B} = AB \sin(0^\circ \text{ or } 180^\circ) \hat{n} = \vec{0}
\]
- Vector product of a vector with itself is a null vector:
\[
\vec{A} \times \vec{A} = A^2 \sin 0^\circ \hat{n} = \vec{0}
\]
- Vector product of orthogonal unit vectors:
\[
\hat{i} \times \hat{j} = \hat{k}, \quad \hat{j} \times \hat{k} = \hat{i}, \quad \hat{k} \times \hat{i} = \hat{j}
\]
\[
\hat{j} \times \hat{i} = -\hat{k}, \quad \hat{k} \times \hat{j} = -\hat{i}, \quad \hat{i} \times \hat{k} = -\hat{j}
\]
6. If:
\[
\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} \\ \quad \text{and} \quad \vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k}
\]
Then:
\[
\vec{A} \times \vec{B}=
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\ \
A_x & A_y & A_z \\ \
B_x & B_y & B_z
\end{vmatrix}
\]
\[
\implies \tiny \vec{A} \times \vec{B} = \hat{i} \left[ A_y B_z – B_y A_z \right] – \hat{j} \left[ A_x B_z – B_x A_z \right] + \hat{k} \left[ A_x B_y – A_y B_x \right]
\]
7. Sine of the angle between two vectors:
\[
\sin \theta = \frac{|\vec{A} \times \vec{B}|}{AB}
\]
8. Unit vector perpendicular to the plane of two vectors:
\[
\hat{n} = \frac{\vec{A} \times \vec{B}}{|\vec{A} \times \vec{B}|}
\]
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