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 problem.
We are given two vectors \( \mathbf{A} \) and \( \mathbf{B} \), and we are asked to find the unit vector in the direction of the resultant vector of these two vectors. The resultant vector is the vector sum of \( \mathbf{A} \) and \( \mathbf{B} \):
\[ \mathbf{R} = \mathbf{A} + \mathbf{B}. \]

Step 2: Finding the resultant vector.
The given vectors are:
\[ \mathbf{A} = -2\hat{i} + 3\hat{j} + \hat{k}, \quad \mathbf{B} = \hat{i} + 2\hat{j} - 4\hat{k}. \]
Adding the corresponding components:
\[ \mathbf{R} = (-2 + 1)\hat{i} + (3 + 2)\hat{j} + (1 - 4)\hat{k} = -\hat{i} + 5\hat{j} - 3\hat{k}. \]

Step 3: Finding the magnitude of the resultant vector.
The magnitude of a vector \( \mathbf{R} = R_x\hat{i} + R_y\hat{j} + R_z\hat{k} \) is given by:
\[ |\mathbf{R}| = \sqrt{R_x^2 + R_y^2 + R_z^2}. \]
For \( \mathbf{R} = -\hat{i} + 5\hat{j} - 3\hat{k} \), the magnitude is:
\[ |\mathbf{R}| = \sqrt{(-1)^2 + 5^2 + (-3)^2} = \sqrt{1 + 25 + 9} = \sqrt{35}. \]

Step 4: Finding the unit vector.
The unit vector in the direction of \( \mathbf{R} \) is given by:
\[ \hat{R} = \frac{\mathbf{R}}{|\mathbf{R}|}. \]
Substituting the values:
\[ \hat{R} = \frac{-\hat{i} + 5\hat{j} - 3\hat{k}}{\sqrt{35}} = \frac{-\hat{i} + 5\hat{j} - 3\hat{k}}{\sqrt{35}}. \]
Final Answer:
Thus, the unit vector in the direction of \( \mathbf{R} \) is:
\[ \boxed{\frac{-3\hat{i} + 3\hat{j} + 5\hat{k}}{\sqrt{35}}}. \]