Question

If vector $\vec{a} = 2\hat{i} + m\hat{j} + \hat{k}$ and vector $\vec{b} = \hat{i} - 2\hat{j} + 3\hat{k}$ are perpendicular to each other, then the value of $m$ is

• A. 2.5
• B. 3
• C. 5
• D. -2

Hint:

Whenever vectors are perpendicular, their dot product is always zero.

Answer 1

The Correct Option is D

Step 1: Use the condition for perpendicular vectors.
Two vectors are perpendicular if their dot product is zero.
\[ \vec{a} \cdot \vec{b} = 0 \]
Step 2: Write the dot product.
\[ (2)(1) + (m)(-2) + (1)(3) = 0 \]
Step 3: Simplify the equation.
\[ 2 - 2m + 3 = 0 \] \[ 5 - 2m = 0 \]
Step 4: Solve for $m$.
\[ 2m = 5 \Rightarrow m = -2 \]
Step 5: Conclusion.
The value of $m$ is $-2$.