Question

Three vectors are expressed as \(\vec{a}=4 \hat{i}-\hat{j}, \vec{b}=-3 \hat{i}+2 \hat{j}\) and \(\vec{c}=-\hat{k}\). The unit vector along the direction of sum of these vectors is

• A. \(\frac{\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}}{\sqrt{3}}\)
• B. \(\frac{1}{3}(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})\)
• C. \(\frac{1}{\sqrt{2}}(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})\)
• D. \(\frac{1}{\sqrt{2}}(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})\)

Hint:

Add the vectors component by component and divide the resultant by its magnitude.

Answer 1

The Correct Option is A

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

Step 2: Add the vectors
\(\vec{R}=(4-3)\hat{i}+(-1+2)\hat{j}-\hat{k}=\hat{i}+\hat{j}-\hat{k}\)

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

Step 4: Compare the options
\(\hat{R}=\dfrac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}\) matches the first option.

Final Answer:
The unit vector is the resultant divided by root 3. \[ \boxed{\text{(A) }\dfrac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}} \]