Vector addition and Vector Subtraction

Vector Addition And Vector Subtraction

Edited By Vishal kumar | Updated on Aug 13, 2024 11:49 AM IST

Vectors are fundamental elements in physics and mathematics, representing quantities that have both magnitude and direction. Unlike scalar quantities, which can be simply added or subtracted using basic arithmetic, vector operations require special consideration of their directional components. Understanding vector addition and subtraction is essential for accurately describing physical phenomena such as forces, velocities, and displacements.

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This article delves into the methods for combining and resolving vectors, starting with the straightforward cases of vectors in the same direction and progressing to more complex scenarios involving different directions, employing the triangle and parallelogram laws of vector addition.

Vector Addition

Vector quantities are not added according to simple algebraic rules, because their direction matters.
The addition of vectors means determining their resultant.
When two vectors are in the same direction then upon addition the direction of the resultant vector is the same as any of the two vectors, while the magnitude of the resultant vector is simply the algebraic sum of two vectors.
eg, Vector $\vec{A}$ has magnitude $4 &$ vector $\vec{B}$ has magnitude 2 in the same direction.

$\vec{A} + \vec{B} = 4 + 2 = 6$

So resultant has a magnitude equal to 6 while its direction is either along $\vec{A}$ or $\vec{B}$

Vector Subtraction

Vector subtraction of $\vec{B}$ from $\vec{A}$ is equal to Vector addition of $\vec{A}$ and negative vector of $\vec{B}$ .
$\vec{A} - \vec{B} = \vec{A} + (- \vec{B})$
E.g., Vector $\vec{A}$ and $\vec{B}$ are in the east direction with magnitudes 4 and 2 respectively.

Vector subtraction of $\vec{B}$ from $\vec{A}$ is equal
$= \vec{A} - \vec{B} = 4 - 2 = 2$

The resultant vector has a magnitude of 2 in the east direction.

Now, let’s understand the different scenarios when both vectors do not have the same direction then the triangle law of vector addition and the Parallelogram law of vector addition come into play.

Triangle law of Vector Addition

If two vectors are represented by both magnitude and direction by two sides of a triangle taken in the same order then their resultant is represented by side of the triangle.

The figure represents the triangle law of vector addition So, the resultant side C is given by

$c = \sqrt{a^{2} + b^{2} + 2 a b \cos θ}$

Where $θ =$ angle between two vectors.

Parallelogram law of Vector Addition

If two vectors are represented by both magnitude and direction by two adjacent sides of a parallelogram taken from the same point then their resultant is also represented by both magnitude and direction taken from the same point but by the diagonal of the parallelogram.

The figure represents the law of parallelogram Vector Addition

Commutative law

The Sum of vectors remains the same in whatever order they may be added.

$\vec{P} + \vec{Q} = \vec{Q} + \vec{P}$

Fig shows the Commutative law of vector addition.

Solved Example Based On Vector Addition And Vector Subtraction

For More Information On Vector Addition And Vector Subtraction, Watch The Below Video:

Now let’s understand the above law by some solved examples.

Example 1: The resultant of two forces 3P and 2P is R. If the first force is doubled, then the resultant is also doubled. The angle between the forces is

1) $120^{\circ}$
2) $60^{\circ}$
3) $180^{\circ}$
4) $90^{\circ}$

Solution:

From the triangle law of vector addition

So using the concept,

$\begin{aligned} R^{2} = (3 P)^{2} + (2 P)^{2} + 2 \cdot (3 P) \cdot (2 P) \cdot \cos θ \\ R^{2} = 13 P^{2} + 12 P^{2} \cos θ . . . . . . . (1) \end{aligned}$

When the first force is doubled, the resultant is doubled
$\begin{aligned} So, (2 R)^{2} = (6 P)^{2} + (2 P)^{2} + 2 \cdot (6 P) (2 P) \cdot \cos θ \\ \Rightarrow 4 R^{2} = 36 P^{2} + 4 P^{2} + 24 P^{2} \cos θ \\ \Rightarrow R^{2} = 10 P^{2} + 6 P^{2} \cos θ . . . . . . . . . . . . (2) \end{aligned}$

Equating (1) and (2)
$\begin{aligned} \Rightarrow 13 P^{2} + 12 P^{2} \cos θ = 10 P^{2} + 6 P^{2} \cos θ \\ \Rightarrow 3 P^{2} = - 6 P^{2} \cos θ \\ \Rightarrow \cos θ = - \frac{1}{2} or θ = 120^{\circ} \end{aligned}$

Equating (1) and (2)

$\begin{aligned} \Rightarrow 13 P^{2} + 12 P^{2} \cos θ = 10 P^{2} + 6 P^{2} \cos θ \\ \Rightarrow 3 P^{2} = - 6 P^{2} \cos θ \\ \Rightarrow \cos θ = - \frac{1}{2} or θ = 120^{\circ} \end{aligned}$

Hence, the answer is the option (1).

Example 2: The ratio of maxium and minimum magnitudes of the resultant of two vectors \text { }|\vec{a}| \text { and }|\vec{b}| is 3:1, Now |\vec{a}|=

∣ \vec{b}

2) (

2 ∣ \vec{b}

3 ∣ \vec{b}

4 | \vec{b} |

Solution:

$\begin{aligned} | \vec{a} + \vec{b} | = \sqrt{a^{2} + b^{2} + 2 a b \cos θ} \\ | \vec{a} + \vec{b} |_{max} = a + b_{when θ = 0^{\circ}} \\ | \vec{a} + \vec{b} |_{min} = a - b when θ = 180^{\circ} \end{aligned}$
here
$\frac{a + b}{a - b} = \frac{3}{1}$
or
$a + b = 3 a - 3 b$
or $2 a = 4 b \Rightarrow a = 2 b$
$| \vec{a} | = 2 | \vec{b} |$

Hence, the answer is option (2).

Example 3:

Two vectors $\vec{A}$ and $\vec{B}$ have equal magnitudes. The magnitudes of $(\vec{A} + \vec{B})$ is 'n' times the magnitudes of $(\vec{A} - \vec{B})$ . The angle between $\vec{A}$ and $\vec{B}$ is :

\sin^{- 1} [\frac{n - 1}{n + 1}]

\cos^{- 1} [\frac{n^{2} - 1}{n^{2} + 1}]

\sin^{- 1} [\frac{n^{2} - 1}{n^{2} + 1}]

\cos^{- 1} [\frac{n - 1}{n + 1}]

Solution:

$\begin{aligned} A^{2} + B^{2} + 2 A B \cos θ = n^{2} (A^{2} + B^{2} - 2 A B \cos θ) \\ ∵ | \vec{A} | = | \vec{B} | \\ ∴ A^{2} + A^{2} + 2 A^{2} \cos θ = n^{2} (A^{2} + A^{2} - 2 A^{2} \cos θ) \\ 2 A^{2} (1 + \cos θ) = 2 A^{2} n^{2} (1 - \cos θ) \\ \cos θ (1 + n^{2}) = n^{2} - 1 \\ \cos θ = \frac{n^{2} - 1}{n^{2} + 1} \\ θ = \cos^{- 1} (\frac{n^{2} - 1}{n^{2} + 1}) \end{aligned}$

Hence, the answer is the Option (2)

Example 4:

The magnitude and direction of the resultant of two vectors $\vec{A}$ and $\vec{B}$ in terms of their magnitudes and angle $θ$ between them is :

R = \sqrt{A^{2} + B^{2} - 2 A B \cos θ}, \tan α = \frac{B \sin θ}{A + B \cos θ}

R = \sqrt{A^{2} + B^{2} + 2 A B \cos θ}, \tan α = \frac{B \cos θ}{A + B \sin θ}

R = \sqrt{A^{2} + B^{2} - 2 A B \cos θ}, \tan α = \frac{A \sin θ}{B + A \sin θ}

R = \sqrt{A^{2} + B^{2} + 2 A B \cos θ}, \tan α = \frac{B \sin θ}{A + B \cos θ}

Solution:

In the Parallelogram law of vector addition

The figure given below represents the law of parallelogram vector Addition.

$\vec{R} = \vec{A} + \vec{B}$

Using Pythagorean theorem in triangle MOP-

$\begin{aligned} R^{2} = (B \sin Θ)^{2} + (A + B \cos Θ)^{2} \\ \Rightarrow & R = \sqrt{A^{2} + B^{2} + 2 A B \cos Θ} \\ \Rightarrow & \tan α = \frac{B \sin Θ}{A + B \cos Θ} \end{aligned}$
Hence, the answer is option (4).

Example 5: Two forces P and Q, of magnitude 2F and 3F, respectively, are at an angle $\theta$ with each other. If the force Q is doubled, then their resultant also gets doubled. Then the angle $\theta$ (in degrees) is :

1) 0

2) 120

3) 90

4) 110

Solution:

Parallelogram law of vector addition -

- wherein

$R^{2} = P^{2} + Q^{2} + 2 P Q \cos θ$

Represents the law of parallelogram Vector Addition
Case 1
$R^{2} = 4 F^{2} + 9 F^{2} + 12 F^{2} \cos θ$

Case 2
$\begin{aligned} 4 R^{2} = 4 F^{2} + 36 F^{2} + 24 F^{2} \cos θ \\ 16 F^{2} + 36 F^{2} + 48 F^{2} \cos θ = 4 F^{2} + 36 F^{2} + 24 F^{2} \cos θ \\ ∴ 12 F^{2} = - 24 F^{2} \cos θ \\ θ = 120^{\circ} \end{aligned}$
Hence, the answer is option (2).

Summary

Understanding vector addition and subtraction is crucial for accurately analyzing physical phenomena involving directional quantities. By mastering the methods for combining vectors, whether in the same or different directions, one can effectively apply these concepts to solve complex problems in physics and engineering, utilizing the triangle and parallelogram laws of vector addition

Frequently Asked Questions (FAQs)

1. What is a vector in physics?

A vector in physics is a quantity that has both magnitude and direction, represented by an arrow.

2. What is a unit vector?

A unit vector is a vector with a magnitude of one. It indicates direction only.

3. How do you multiply vectors?

Vectors can be multiplied in several ways: scalar multiplication (multiplying a vector by a scalar quantity), dot product (or scalar product, yielding a scalar), and cross product (or vector product, yielding another vector).

4. When do you use vector components?

Vector components are used to simplify vector calculations, especially in coordinate systems like Cartesian coordinates. They break down a vector into its perpendicular components along the coordinate axes (x, y, z).