Q205 / 270 · NDA 2013 (Phase 2) · Mathematics

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

  • Correct answer
    A
    8
  • B
    6
  • C
    4
  • D
    2

Solution

$\begin{aligned} \vec{a}+t \vec{b} & =(2 \hat{i}+2 \hat{j}+3 \hat{k})+t(-\hat{i}+2 \hat{j}+\hat{k}) \\ & =(2-t) \hat{i}+(2+2 t) \hat{j}+(3+t) \hat{k} \end{aligned}$ According to the question, $(\vec{a}+t \vec{b}) \cdot \vec{c}=0$ $(2-t) \cdot 3+(2+2 t) \cdot 1=0$ $8-t=0 \Rightarrow t=8$

Hence, the correct answer is option: 'A'.

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