Question

$P\equiv \left(x_1,y_1,z_1\right)$ and $Q\equiv \left(x_2,y_2,z_2\right)$ are two points if direction cosines of a line $AB$ are $l,m,n$ then the projection of $PQ$ on $AB$ are-

• A. $l(x_2-x_1)+m(y_2-y_1)+n(z_2-z_1)$
• B. $l(x_2-x_1)-m(y_2-y_1)-n(z_2-z_1)$
• C. $0$
• D. None of these

Answer 1

The correct option is A $l(x_2-x_1)+m(y_2-y_1)+n(z_2-z_1)$

$\overrightarrow{PQ}=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}$
$\text{unit vector along AB}=\hat{n}=l\hat{i}+m\hat{j}+n\hat{k}$

Projection of $\overrightarrow{PQ}$ on $\overrightarrow{AB}=\overrightarrow{PQ}\cdot \hat{n}=l(x_2-x_1)+m(y_2-y_1)+n(z_2-z_1)$