Question

If $\overrightarrow{a}=2\hat{i}+2\hat{j}+3\hat{k}$, $\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}$ and $\overrightarrow{c}=3\hat{i}+\hat{j}$ are three vectors such that $\overrightarrow{a}+t\overrightarrow{b}$ is perpendicular to $\overrightarrow{c}$, then what is $t$ equal to?

• A. $8$
• B. $6$
• C. $4$
• D. $2$

Answer 1

Answer: (A)

$8$

Given vectors

$\overrightarrow{a}=2\hat{i}+2\hat{j}+3\hat{k}$

$\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}$

$\overrightarrow{c}=3\hat{i}+\hat{j}$

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

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

$(\overrightarrow{a}+t\overrightarrow{b})\cdot \overrightarrow{c}=0$

$\left(\right(2-t)\hat{i}+(2+2t)\hat{j}+(3+t\left)\hat{k}\right)\cdot (3\hat{i}+\hat{j})=0$

$3(2-t)+(2+2t)=0$

$6-3t+2+2t=0$

$8-t=0$

$t=8$