Class 11 > Unit # 02: Kinematics > Addition of Vectors by Rectangular Component Method

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Unit # 2: Kinematics — Rectangular Components Method for Vector Addition

Lecture Overview

This lecture from Talha's Physics Academy provides a comprehensive guide on adding vectors using their rectangular components in Class 11 Physics (Unit 2: Kinematics), covering component resolution, resultant magnitude, and direction formulas.

Source Video: Rectangular Components Method | Unit # 2 Kinematics | Class 11 Physics

Rectangular Components Method

The way of adding vectors with the help of their rectangular components is called addition of vectors by rectangular components method.

Suppose two position vectors \(\vec{v}_1\) and \(\vec{v}_2\) having lengths or magnitudes \(V_1\) and \(V_2\) and making angles \(\theta_1\) and \(\theta_2\) respectively are to be added.

For this purpose we first adopt head-to-tail rule and then we draw perpendiculars from their heads on x and y axes to get their rectangular components as shown in the figure.

Rectangular Components of Vectors

The rectangular components of \(\vec{v}_1\) are:

$$v_{1x} = v_1 \cos\theta_1$$

$$v_{1y} = v_1 \sin\theta_1$$

The rectangular components of \(\vec{v}_2\) are:

$$v_{2x} = v_2 \cos\theta_2$$

$$v_{2y} = v_2 \sin\theta_2$$

Resultant Components

It is clear from the figure that for the x-component:

$$R_x = v_{1x} + v_{2x}$$

$$R_x = v_1 \cos\theta_1 + v_2 \cos\theta_2$$

Similarly, for the y-component:

$$R_y = v_{1y} + v_{2y}$$

$$R_y = v_1 \sin\theta_1 + v_2 \sin\theta_2$$

Magnitude and Direction of Resultant Vector

Now, the resultant magnitude can be calculated by the formula:

$$R = \sqrt{R_x^2 + R_y^2}$$

$$R = \sqrt{(v_1 \cos\theta_1 + v_2 \cos\theta_2)^2 + (v_1 \sin\theta_1 + v_2 \sin\theta_2)^2}$$

The direction (\(\theta\)) of the resultant vector is given by:

$$\tan\theta = \frac{R_y}{R_x}$$

$$\theta = \tan^{-1}\left(\frac{v_1 \sin\theta_1 + v_2 \sin\theta_2}{v_1 \cos\theta_1 + v_2 \cos\theta_2}\right)$$

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