Question

A unit vector in the direction of resultant vector of \(\vec{A}=-2 \hat{i}+3 \hat{j}+\hat{k}\) and \(\vec{B}=\hat{i}+2 \hat{j}-4 \hat{k}\) is

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

Hint:

Add the vectors component by component, then divide by the magnitude of the resultant.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
A unit vector along a vector \(\vec{R}\) is \(\hat{R}=\frac{\vec{R}}{|\vec{R}|}\).

Step 2: Resultant
\(\vec{R}=\vec{A}+\vec{B}=(-2+1)\hat{i}+(3+2)\hat{j}+(1-4)\hat{k}=-\hat{i}+5\hat{j}-3\hat{k}\).

Step 3: Magnitude
\(|\vec{R}|=\sqrt{1+25+9}=\sqrt{35}\).

Step 4: Compare the options
\(\hat{R}=\frac{-\hat{i}+5\hat{j}-3\hat{k}}{\sqrt{35}}\), option (D).

Final Answer:
The unit vector is \(\frac{-\hat{i}+5\hat{j}-3\hat{k}}{\sqrt{35}}\). \[ \boxed{\text{(D) }\dfrac{-\hat{\mathrm{i}}+5\hat{\mathrm{j}}-3\hat{\mathrm{k}}}{\sqrt{35}}} \]