Question

lf the projections ofthe line segment$AB$ on the $yz$-plane, $zx$-plane, $xy$-plane are $\sqrt{160},\sqrt{153},5$ respectively, then the projection of $AB$ on the $z$-axis is

• A. $\sqrt{12}$
• B. $\sqrt{13}$
• C. $12$
• D. $144$

Answer 1

Answer: (C)

$12$

Let $\overrightarrow{AB}=l\hat{i}+m\hat{j}+n\hat{k}$

$l,m,n$ are projections of $\overrightarrow{AB}$ on $x,y,z$ respectively

Projection of $\overrightarrow{AB}$ on $yz$ plane $=\sqrt{m^2+n^2}=\sqrt{160}$

Projection of $\overrightarrow{AB}$ on$zx$ plane $=\sqrt{n^2+l^2}=\sqrt{153}$

Projection of $\overrightarrow{AB}$ on$xy$ plane $=\sqrt{l^2+m^2}=5$

$(m^2+n^2)-(n^2+l^2)=160-153$

$(m-l)(m+l)=7$

$\Rightarrow m=4,l=3,n=12$

Projection of $AB$ on $z-$ axis $=n=12$