Question

A unit vector in the direction of resultant vector of \[ \mathbf{A} = -2\hat{i} + 3\hat{j} + \hat{k}, \quad \mathbf{B} = \hat{i} + 2\hat{j} - 4\hat{k} \] is

• A. \( \frac{-3\hat{i} + 3\hat{j} + 5\hat{k}}{\sqrt{35}} \)
• B. \( \frac{\hat{i} + 2\hat{j} + 4\hat{k}}{\sqrt{35}} \)
• C. \( \frac{-2\hat{i} + 3\hat{j} + \hat{k}}{\sqrt{35}} \)
• D. \( \frac{\hat{i} + 3\hat{j} - 5\hat{k}}{\sqrt{35}} \)

Hint:

To find the unit vector in the direction of a vector, first find the resultant vector, then divide it by its magnitude.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find the resultant vector of \( \vec{A} \) and \( \vec{B} \), and then find its unit vector.
Step 2: Key Formula or Approach:
Resultant vector \( \vec{R} = \vec{A} + \vec{B} \).
Unit vector \( \hat{R} = \frac{\vec{R}}{|\vec{R}|} \).
Step 3: Detailed Explanation:
Calculate the resultant vector:
\[ \vec{R} = (-2+1)\hat{i} + (3+2)\hat{j} + (1-4)\hat{k} \]
\[ \vec{R} = -\hat{i} + 5\hat{j} - 3\hat{k} \]
Calculate the magnitude of \( \vec{R} \):
\[ |\vec{R}| = \sqrt{(-1)^2 + 5^2 + (-3)^2} = \sqrt{1 + 25 + 9} = \sqrt{35} \]
Find the unit vector:
\[ \hat{R} = \frac{-\hat{i} + 5\hat{j} - 3\hat{k}}{\sqrt{35}} \]
Step 4: Final Answer:
The unit vector is \( \frac{-\hat{i} + 5\hat{j} - 3\hat{k}}{\sqrt{35}} \).