Question

At what value of angle between two vectors will the resultant of the two vectors be maximum?

Answer 1

The resultant is maximum when the angle between the two vectors is zero

Explanation:

The formula to calculate the resultant of two vectors A, B is the following:

$R=\sqrt{A^2+B^2+2ABcos \theta}$

where

A is the magnitude of vector A

B is the magnitude of vector B

$\theta$ is the angle between the directions of A and B

This formula can be derived by applying the parallelogram law.

From the formula, we observe that:

- When $\theta=0^{\circ}$ (vectors A and B parallel), $cos \theta = 1$. so the resultant becomes simply

$R=\sqrt{A^2+B^2 + 2AB}=\sqrt{(A+B)^2}=A+B$

- When $\theta = 90^{\circ}$ (vectors A and B perpendicular), $cos \theta = 0$, so the resultant becomes

$R=\sqrt{A^2+B^2}$

- When $\theta=180^{\circ}$ (vectors A and B anti-parallel), $cos \theta = -1$, so the resultant becomes

$R=\sqrt{A^2+B^2 - 2AB}=\sqrt{(A-B)^2}=|A-B|$

By looking at the three cases and at the formula, we see that the maximum value of the resultant is when the angle between the two vectors is zero, since in that case the resultant is simply obtained by adding the magnitudes of the two vectors.

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