Question

$If \, \, \vec{a}=\hat{i}+2\hat{j}+3\hat{k}, \, \vec{b}= \, -\hat{i}+2\hat{j}+\hat{k}$ and $\vec{c} \, = \, 3\hat{i}+\hat{j}$ then t such that $\vec{a}+t\vec{b}$ is at right angle to $\vec{c}$ will be equal to

• A. 5
• B. 4
• C. 6
• D. 2

Answer 1

The Correct Option is A

Two vectors are perpendicular if their dot product is zero.

Given vectors:

\(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \quad \vec{b} = -\hat{i} + 2\hat{j} + \hat{k}, \quad \vec{c} = 3\hat{i} + \hat{j}\)

First form the vector:

\(\vec{a} + t\vec{b} = (1 - t)\hat{i} + (2 + 2t)\hat{j} + (3 + t)\hat{k}\)

Now take dot product with \(\vec{c}\):

\((\vec{a} + t\vec{b}) \cdot \vec{c} = (1 - t)\cdot 3 + (2 + 2t)\cdot 1 + (3 + t)\cdot 0\)

Simplifying:

\(= 3 - 3t + 2 + 2t\)

\(= 5 - t\)

For perpendicularity:

\(5 - t = 0\)

\(t = 5\)

Final answer: \(t = 5\)