Question

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

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

Answer 1

The correct option is C $\frac{6}{\sqrt{14}}$

The line can be as well written as $3\overrightarrow{i}-\overrightarrow{j}+t(\overrightarrow{i}+2\overrightarrow{j}+3\overrightarrow{k})$
Now the projection of $\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}$ on the line is given by the projection onto the direction vector , i.e. $\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{b}|}$ in general is the projection of $\overrightarrow{a}$ on $\overrightarrow{b}$
Here, its $\frac{1+2+3}{\sqrt{1+4+9}}=\frac{6}{14}$