Question

The vectors $2\hat{i}+3\hat{j}-4\hat{k}$ and $a\hat{i}+b\hat{j}+c\hat{k}$ are perpendicular if
(a) a = 2, b = 3, c = −4
(b) a = 4, b = 4, c = 5
(c) a = 4, b = 4, c = −5
(d) a = −4, b = 4, c = −5

Answer 1

(b) a = 4, b = 4, c = 5

$$\begin{gathered} \text{It is given that vectors 2}\hat{i}+3\hat{j}-4\hat{k}\mathrm{and}a\hat{i}+b\hat{j}+c\hat{k}\text{are perpendicular.} \\ \text{So, their dot product is zero.} \\ \Rightarrow 2a+3b-4c=0 \\ \left(b\right)a=4;b=4;c=5 \\ \Rightarrow 2\left(4\right)+3\left(4\right)-4\left(5\right)=0 \\ 8+12-20=0 \\ 0=0,\text{which is true.} \end{gathered}$$