Question

The projection of the vector $\hat{i}+\hat{j}+\hat{k}$ on the line whose vector equation is $\overrightarrow{r}= (3+t)\hat{i}+(2t-1)\hat{j}+(3t)\hat{k}, t$ being a scalar , is

• A. $\frac{1}{\sqrt{14}}$
• B. 6
• C. $\frac{6}{\sqrt{14}}$
• D. none of these

Answer 1

Answer: (C)

$\vec{r}=3\,\hat{i}-\hat{j}+t\left(\hat{i}+2\,\hat{j}+3\,\hat{k}\right)$
$\therefore $ a vector parallel to the given line is
$\vec{b}=\hat{i}+2\,\hat{j}+3\,\hat{k}$
$\therefore $ unit vector along the line
$=\frac{\hat{i}+2\,\hat{j}+3\,\hat{k}}{\sqrt{1+4+9}}=\frac{\hat{i}+2\,\hat{j}+3\,\hat{k}}{\sqrt{14}}$
$\therefore $ projection $=\left(\hat{i}+2\,\hat{j}+3\,\hat{k}\right)\cdot \frac{\left(\hat{i}+2\,\hat{j}+3\,\hat{k}\right)}{\sqrt{14}}$
$=\frac{1+2+3}{\sqrt{14}}=\frac{6}{\sqrt{14}}$