Can three vectors of equal magnitude ever result in a zero resultant?

Yes. If the three vectors are coplanar and their directions are such that they form an equilateral triangle (i.e., each vector is at 120 degrees to the other two), their resultant will be zero.

Is vector addition commutative? Give an example.

Yes, vector addition is commutative. This means that A + B = B + A. Imagine pushing a box first horizontally and then vertically. The final displacement is the same if you had pushed it vertically first and then horizontally.

What is the physical significance of the cross product of two vectors?

The cross product (A x B) produces a vector perpendicular to both A and B, with magnitude equal to the parallelogram's area. Its direction follows the right-hand rule and is used in torque, angular momentum, and force on a moving charge in a magnetic field.

If the cross product of two non-zero vectors is zero, what can you conclude about the angle between them?

The vectors are parallel or anti-parallel, with an angle of 0° or 180°. Since |A x B| = |A||B|sin θ, if the cross product is zero and neither vector is zero, then sin θ = 0, so θ must be 0° or 180°.

Can a vector have zero magnitude but non-zero components?

No. If a vector has zero magnitude, all its components must be zero. The magnitude of a vector is determined using the Pythagorean theorem, whether in 2D or 3D. Therefore, if the magnitude is zero, all the terms inside the square root must also be zero.

Understanding Vectors in Physics: Key Concepts & Applications

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JEE Physics

Vectors in Physics

Vectors are quantities that have both size (magnitude) and direction, like velocity and force. They help us describe and solve problems related to motion, forces, and other physical phenomena. Essentially, vectors allow us to represent real-world situations that involve movement or influence, making them key to understanding and analyzing how things move or interact in the physical world.

1.0Definition of Vector Quantities

If a physical quantity has both magnitude and direction,

(a) It obeys commutative law of additions: $A + B = B + A$

(b) It obeys the rule of vector algebra.

Example-Displacement, velocity, acceleration, force etc.

Representation of Vector

A vector is represented by a line headed with an arrow.
Its length is proportional to its magnitude and the arrow indicates direction.

$A i s aV ec t or, A = PQ, M a g ni t u d eo f A i s A or A$

Important Points:

If a vector is shifted along its own direction, it remains unchanged.

If a vector is rotated by an angle that is not a multiple of 360 degrees, its direction will change of $2 π (or 360°)$ it changes.

When the frame of reference is shifted or rotated, the vector itself remains unchanged, although its components may vary.

The angle between two vectors refers to the smaller of the two angles formed when the vectors are placed tail-to-tail or head-to-head, with one vector displaced parallel to itself. i.e. $0 \leq θ \leq π$

2.0Types of Vector

1.Parallel Vectors:Those vectors which have the same direction are called parallel vectors.The angle between two parallel vectors is always 0°.

2.Equal Vectors:Vectors which have equal magnitude, same direction and they represent the same physical quantity are called equal vectors.

$A = B$

3.Anti–Parallel Vectors:Those vectors which have opposite direction are called anti–parallel vectors.Angle between two anti–parallel vectors is always 180°.

4.Negative (or Opposite) Vectors:Vectors which have equal magnitude but opposite direction are called negative vectors of each other.

$A B an d B A a re n e g a t i v e v ec t ors A B = - B A$

5.Co-Initial Vectors:Co-initial vectors are those vectors which have the same initial point.In figure $a, b, c$ , are co-initial vectors.

6.Collinear Vectors: The vectors lying in the same line are known as collinear vectors.

The angle between collinear vectors is either 0° or 180°.

Example:

$(1) \leftarrow\leftarrow (θ = 0°) (2) \to\to (θ = 0°) (3) \leftarrow\to (θ = 180°) (4) \to\leftarrow (θ = 180°)$

7.Coplanar Vectors: Vectors located in the same plane are called coplanar vectors.

Two vectors are always coplanar.

8.Concurrent Vectors: Those vectors which pass through a common point are called concurrent vectors.

In figure $a, b, c$ are concurrent vectors.

9.Null or Zero Vector: A vector having zero magnitude is called a null vector. The direction of a null vector is indeterminate.

Sum of two vectors is always a vector so, $(A) + (- A) = 0$ .

$0$ is a zero vector or null vector.

10.Unit Vector:A vector having unit magnitude is called unit vector. It is used to specify direction. A unit vector is represented by $A$ (Read as A cap or A hat or A caret).

Unit vector in the direction of $A i s, A = \frac{A}{A}$

$U ni t v ec t or = \frac{V ec t or}{M o d u l u s o f t h e V ec t or}$

A unit vector is used to specify the direction of a vector.

Base Vectors:In an XYZ coordinate frame there are three unit vectors $i, j an d k$ These are used to indicate X, Y and Z directions respectively and are called as base vectors.These three unit vectors are mutually perpendicular to each other.

Multiplication of a Vector by a Scalar

Multiplying a vector by a scalar alters its magnitude, but the direction remains the same (if the scalar is positive) or reverses (if the scalar is negative).
If, $B = 2 A$ ,

If, $C = \frac{A}{2}$

If, $D = - A$ ,

3.0Addition of Two Vectors

1.Triangle Law of Addition of Two Vectors:If two vectors form two sides of a triangle, their sum is given by the third side, taken in the opposite direction.

Triangle Law of Addition of Two Vectors A

$R es u lt an t R = A + B$

Triangle Law of Addition of Two Vectors B

$R = A + B = A^{2} + B^{2} + 2 A B C os θ$

$θ$ is the angle between $A an d B$

Let direction of $R mak e α an g l e w i t h A T an α = \frac{B S in θ}{A + BC os θ}$

2.Parallelogram Law of Addition of Two Vectors:If two vectors form adjacent sides of a parallelogram, their resultant is represented by the diagonal passing through their common point.

$A B + A D = A C \Rightarrow A + B = R$

$R = A^{2} + B^{2} + 2 A B C os θ$

$T an α = \frac{BS in θ}{A + B C os θ}$

$T an β = \frac{A S in θ}{A + B C os θ}$

4.0Vectors (Law of Polygon)

If some vectors are represented by sides of a polygon in the same order, then their resultant vector is represented by the closing side of the polygon in the opposite order.

$R = A + B + C + D$

In a polygon if all the vectors taken in the same order are such that the head of the last vector coincides with the tail of the first vector then their resultant is a null vector.

If n coplanar vectors of equal magnitude are arranged at equal angles of separation then their resultant is always zero.

Important Points

Vector addition is commutative $A + B = B + A$
Vector addition is associative $A + (B + C) = (A + B) + C$
Vectors representing the same physical quantity only, can be added.
The resultant of two vectors will be maximum when they are parallel i.e. angle between them is zero. $R_{ma x} = A + B$
The resultant of two vectors will be minimum when they are antiparallel i.e. angle between them is 180°. $R_{m i n} = A - B$
Three non-coplanar vectors cannot have a zero resultant; at least four are needed for this.
If two vectors have equal magnitude i.e. $A = B = a$ and angle between them is then resultant will be along the angle bisector of $A an d B$ and its magnitude is equal to $2 a C os \frac{θ}{2}$

$R = A + B = 2 a C os \frac{θ}{2}$

5.0Subtraction of Two Vectors

Let $A an d B$ are two vectors. Their difference i.e. $A - B$ can be treated as sum of the vector $A an d v ec t or (- B), A - B = A + (- B)$

$A - B = A^{2} + B^{2} - 2 A B C os θ, T an α = \frac{B S in θ}{A - BC os θ}$

Key Points:

Vector subtraction does not follow commutative law i.e. $A - B \neq = B - A$
Vector subtraction does not follow associative law i.e $A - B - C \neq = A - B - C$
If two vectors have equal magnitude, i.e. $A = B = α$ and $θ$ is the angle between them then $A - B = 2 a S in \frac{θ}{2}$
The change in a vector quantity is the difference between the final and initial vectors.

The parallelogram formed by two vectors $A an d B$ will have two diagonals, one diagonal represents

$A + B an d an o t h er d ia g o na l re p rese n t s A - B$

Components of Vector:A vector can be expressed as a vector sum of two or more than two vectors, then these vectors are known as components of vectors.

Non-rectangular Components	Rectangular Component

$(a an d b$ are non-rectangular components.)	( $a an d b$ are non-rectangular components.)
R=A+B	R=A+B ,If the vectors are perpendicular to each other or the angle between the component vector is 90°, then it is known rectangular components

6.0Resolution of Vectors

1.Resolution of Vectors into Rectangular Components in Two Dimensions

When a vector is split into components which are at right angles to each other, then the components are called rectangular or orthogonal components of that vector.

$a_{x} = a C os α a_{y} = a S in α$

$a = a_{x}^{2} + a_{y}^{2} T an = \frac{a _{y}}{a _{x}}$

Rectangular Components of a Vector in(3D)

$O A = O B + O D + O C$

$a = a_{x} i + a_{y} j + a_{z} k$

$a = a_{x}^{2} + a_{y}^{2} + a_{z}^{2}$

Direction Cosines

$C os α = \frac{a _{x}}{a} C os β = \frac{a _{y}}{a} C os γ = \frac{a _{z}}{a}$

$C o s^{2} α + C o s^{2} β + C o s^{2} γ = 1$

$S i n^{2} α + S i n^{2} β + S i n^{2} γ = 2$

Addition & Subtraction in Component Form

$A = A_{x} i + A_{y} j + A_{z} k$

$B = B_{x} i + B_{y} j + B_{z} k$

$A \pm B = (A_{x} \pm B_{x}) i + (A_{y} \pm B_{y}) j + (A_{z} \pm B_{z}) k$

$C = C_{x} i + C_{y} j + C_{z} k$

$A \mp B \mp C = (A_{x} \mp B_{x} \mp C_{x}) i + (A_{y} \mp B_{y} \mp C_{y}) j + (A_{z} \mp B_{z} \mp C_{z}) k$

Modulus of vector $A, A = A_{x}^{2} + A_{y}^{2} + A_{z}^{2}$

Position vector (P.V.) of a point

$OP = x \hat{i} + y \hat{j} + z \hat{k}$

Displacement vector:Change in position vector is known as displacement vector.

$r_{i} = x_{1} \hat{i} + y_{1} \hat{j} + z_{1} \hat{k} and r_{f} = x_{2} \hat{i} + y_{2} \hat{j} + z_{2} \hat{k}$ Displacement Vector $(s) = r_{f} - r_{i} = (x_{2} \hat{i} + y_{2} \hat{j} + z_{2} \hat{k}) - (x_{1} \hat{i} + y_{1} \hat{j} + z_{1} \hat{k})$ $(s) = (x_{2} - x_{1}) \hat{i} + (y_{2} - y_{1}) \hat{j} + (z_{2} - z_{1}) \hat{k}$

7.0Scalar Product

The scalar product, or dot product, of two vectors is a mathematical operation that multiplies their magnitudes and the cosine of the angle between them. It results in a scalar value and is denoted as A ⋅ B = |A| |B| cos(θ), where A and B are the vectors, and θ is the angle between them.

Properties of Scalar Product

1.The scalar product is always a scalar value. It is positive when the angle between the vectors is acute (less than 90°) and negative when the angle is obtuse (between 90° and 180°).

2. It is Commutative $A \cdot B = B \cdot A$ 3. It is Distributive $A \cdot (B + C) = A \cdot B + A \cdot C)$

4.By Definition, $A \cdot B = A B cos θ \Rightarrow θ = cos^{- 1} [\frac{A \cdot B}{A B}]$

5.Geometrically, B cos(θ) represents the projection of B onto A, while A cos(θ) represents the projection of A onto B.

6.Scalar product of two vectors will be maximum $cos θ = max = + 1 i . e . θ = 0^{\circ}$ vectors are parallel, $(A \cdot B)_{max} = A B$

7.If the scalar product of two non zero vectors is zero then the vectors are perpendicular.

8.The scalar product of a vector by itself is termed as self dot product

$(A)^{2} = A \cdot A = AA cos θ = AA cos 0^{\circ} = A^{2} \Rightarrow A = A \cdot A$

9.In case of unit vector, $\overset{n}{^} \cdot \overset{n}{^} = 1 \times 1 \times cos 0^{\circ} = 1 \Rightarrow \overset{n}{^} \cdot \overset{n}{^} = \hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$

10.In case of orthogonal unit vectors $\hat{i}, \hat{j} an d \hat{k} (\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0)$

$A \cdot B = (A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}) \cdot (B_{x} \hat{i} + B_{y} \hat{j} + B_{z} \hat{k}) = [A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}]$

Projection of $A o n B$

(a) In scalar form:Projection of $A on B = A cos θ = A (\frac{A \cdot B}{A B}) = \frac{A \cdot B}{B} = A \cdot \hat{B}$

(b) In vector form :Projection of $A on B = A cos θ \hat{B} = (\frac{A \cdot B}{B}) \hat{B} = (A \cdot \hat{B}) \hat{B}$

8.0Vector Product

The vector product or cross product of two vectors $A an d B$ ,denoted as

$A \times B (re a d A cross B)$ .

$A \times B = A B S in θ n$ .

Here \theta is the angle between the vectors and the direction is given by the right-hand-thumb rule.

Properties of Vector Product

The cross product of two vectors is always a vector perpendicular to the plane formed by the two vectors.

2.Vector product of two vectors is not commutative $A \times B = \neq = B \times A$

3.The vector product is distributive when the order of the vectors is strictly maintained

$A \times (B + C) = A \times B + A \times C$

4.The magnitude of vector product of two vectors will be maximum when $S in θ_{ma x} = 1, i . e = 90°$

$A \times B_{ma x} = A B$ , The magnitude of the cross product is greatest when the vectors are perpendicular.

5.The magnitude of the cross product of two non-zero vectors will be at its minimum,

$sin θ = minimum = 0$ ,

If the cross product of two non-zero vectors is zero, the vectors are collinear.

Either $θ = 0^{\circ} or 18 0^{\circ} or ∣ A \times B ∣_{min} = 0$

6.The cross product of a vector with itself equals zero. i.e. is a null vector.

$A \times A = AA sin 0^{\circ} = 0$

$\overset{n}{^} \times \overset{n}{^} = 0$

$\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$

7. In case of orthogonal unit vectors $\hat{i}, \hat{j} and \hat{k}$ in compliance with Right Hand Thumb Rule,

$\hat{i} \times \hat{j} = \hat{k}$

$\hat{j} \times \hat{k} = \hat{i}$

$\hat{k} \times \hat{i} = \hat{j}$

8.In terms of components,

$A \times B = \hat{i} A_{x} B_{x} \hat{j} A_{y} B_{y} \hat{k} A_{z} B_{z} = \hat{i} A_{y} B_{y} A_{z} B_{z} - \hat{j} A_{x} B_{x} A_{z} B_{z} + \hat{k} A_{x} B_{x} A_{y} B_{y}$

$A \times B = \hat{i} (A_{y} B_{z} - A_{z} B_{y}) + \hat{j} (A_{z} B_{x} - A_{x} B_{z}) + \hat{k} (A_{x} B_{y} - A_{y} B_{x})$

Examples of Vector Product

(1) Torque $(τ = r \times F)$

(2) Velocity $(v = ω \times r)$

Formulae to Find Area:

If $A and B$ are two abutting sides of a triangle, then its area, $= \frac{1}{2} ∣ A \times B ∣$
If $A and B$ are two adjacent sides of a parallelogram, then its area, $= ∣ A \times B ∣$
If $A and B$ are diagonals of a parallelogram,then its area, $= \frac{1}{2} ∣ A \times B ∣$

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