Question

Two vectors are given by $\vec{A} = \hat{i} - 2\hat{j} + 3\hat{k}$ and $\vec{B} = 2\hat{i} - 4\hat{j} + 3\hat{k}$. The unit vector along $\vec{A} + \vec{B}$ is

• A. $3\hat{i} - 6\hat{j} + 6\hat{k}$ (Note: Option incomplete/incorrect in source, calculating correct value)
• B. $\frac{\hat{i} - 2\hat{j} + 2\hat{k}}{3}$
• C. $\frac{\hat{i} + 2\hat{j} + 2\hat{k}}{7}$
• D. $\frac{3\hat{i} + 4\hat{j} + 5\hat{k}}{3}$

Hint:

To find a unit vector, simply divide the vector by its own magnitude (length). $\hat{n} = \frac{\vec{V}}{|\vec{V}|}$.

Answer 1

The Correct Option is B

Step 1: Vector Addition:
First, we find the resultant vector $\vec{R} = \vec{A} + \vec{B}$. Given: \[ \vec{A} = \hat{i} - 2\hat{j} + 3\hat{k} \] \[ \vec{B} = 2\hat{i} - 4\hat{j} + 3\hat{k} \] Adding corresponding components: \[ \vec{R} = (1+2)\hat{i} + (-2-4)\hat{j} + (3+3)\hat{k} \] \[ \vec{R} = 3\hat{i} - 6\hat{j} + 6\hat{k} \]
Step 2: Magnitude of Resultant Vector:
The magnitude $|\vec{R}|$ is given by $\sqrt{R_x^2 + R_y^2 + R_z^2}$. \[ |\vec{R}| = \sqrt{3^2 + (-6)^2 + 6^2} \] \[ |\vec{R}| = \sqrt{9 + 36 + 36} = \sqrt{81} = 9 \]
Step 3: Calculating the Unit Vector:
The unit vector $\hat{n}$ along $\vec{R}$ is defined as $\frac{\vec{R}}{|\vec{R}|}$. \[ \hat{n} = \frac{3\hat{i} - 6\hat{j} + 6\hat{k}}{9} \] We can simplify this by factoring out $3$ from the numerator: \[ \hat{n} = \frac{3(\hat{i} - 2\hat{j} + 2\hat{k})}{9} \] \[ \hat{n} = \frac{\hat{i} - 2\hat{j} + 2\hat{k}}{3} \]
Step 4: Final Answer:
The unit vector is $\frac{\hat{i} - 2\hat{j} + 2\hat{k}}{3}$.