Q.If A$(1,2,3)$ and B$(4,5,6)$ are two points, then find the foot of the perpendicular from the point B to the line joining the origin and point A.

$O=(0,0,0),A=(1,2,3),B=(4,5,6)$. The line joining $O$ and $A$ has direction vector

$\overrightarrow{OA}=\hat i+2\hat j+3\hat k$. The foot of the perpendicular from $B$ to this line is the

vector projection of $\overrightarrow{OB}=4\hat i+5\hat j+6\hat k$ onto $\overrightarrow{OA}$:

$\overrightarrow{OB}\cdot\overrightarrow{OA}=4(1)+5(2)+6(3)=4+10+18=32$. $|\overrightarrow{OA}|^2=1+4+9=14$. …

Recognise that the foot of a perpendicular from a point to a line through the origin is exactly the vector projection of that point's position vector …