Question

The magnitude of vectors $ \overrightarrow{A}$, $\overrightarrow{B}$ and $ \overrightarrow{C} $ are 3, 4 and 5 units respectively. If $ \overrightarrow{A} + \overrightarrow{B} = \overrightarrow{C}$, the angle between $ \overrightarrow{A} $ and $ \overrightarrow{B} $ is

• A. $\pi / 2$
• B. $ \cos^{ - 1} (0 . 6)$
• C. $ \tan^{ - 1} \, (7/5)$
• D. $ \pi / 4 $

Answer 1

The Correct Option is A

To solve this problem, we need to understand the geometrical arrangement of the vectors and how vector addition works.

Given:

• Magnitudes of vectors: $|\overrightarrow{A}| = 3$, $|\overrightarrow{B}| = 4$, $|\overrightarrow{C}| = 5$.
• Vector equation: $\overrightarrow{A} + \overrightarrow{B} = \overrightarrow{C}$.

The vector equation $\overrightarrow{A} + \overrightarrow{B} = \overrightarrow{C}$ implies that vectors $ \overrightarrow{A} $, $ \overrightarrow{B} $, and $ \overrightarrow{C} $ form a right triangle with $ \overrightarrow{C} $ as the hypotenuse.

In a right-angled triangle formed by vectors, the relation according to the Pythagorean theorem is:

$$|\overrightarrow{C}|^2 = |\overrightarrow{A}|^2 + |\overrightarrow{B}|^2.$$

Substituting the given magnitudes:

• $5^2 = 3^2 + 4^2$.
• $25 = 9 + 16$.
• $25 = 25$, which confirms the vector relation is correct.

This confirms that the angle between $ \overrightarrow{A} $ and $ \overrightarrow{B} $, denoted by $ \theta $, is $\pi/2$ (90 degrees), which represents a right angle.

Therefore, the angle between $ \overrightarrow{A} $ and $ \overrightarrow{B} $ is $\pi/2$.

Justification for incorrect options:

• $ \cos^{-1}(0.6) $, $ \tan^{-1}(7/5) $, and $ \pi/4 $ do not satisfy the vector conditions as calculated.

Hence, the correct option is $\pi/2$.