Question

Let $ABCD$ be the parallelogram whose sides $AB$ and $AD$ are represented by the vectors $2\hat{i}+4\hat{j}-5\hat{k}$ and $\hat{i}+2\hat{j}+3\hat{k}$ respectively. Then, if a is a unit vector parallel to $AC$, then $a$ is equal to

• A. $(3\hat{i}-6\hat{j}-2\hat{k})/3$
• B. $(3i+6\hat{j}+2\hat{k})/3$
• C. $(3\hat{i}-6\hat{j}-3\hat{k})/7$
• D. $(3\hat{i}+6\hat{j}-2\hat{k})/7$

Answer 1

Answer: (D)

Let $R_1=2\hat{i}+4\hat{j}-5\hat{k}$

and $R_2=\hat{i}+2\hat{j}+3\hat{k}$
$\therefore$ R(along AC) $=R_1+R_2$
$=3\hat{i}+6\hat{j}-2\hat{k}$
$\therefore$ a (unit vector along A C) $=\frac{R}{|R|}$
$=\frac{3\hat{i}+6\hat{j}-2\hat{k}}{\sqrt{\hat{9}+3\hat{6}+1}}$
$=\frac{1}{7}(3\hat{i}+6\hat{j}-2\hat{k})$