Question

lf the projections of the 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. $2$
• B. $6$
• C. $8$
• D. $12$

Answer 1

The correct option is D $12$

Let $AB=l\hat{i}+m\hat{j}+n\hat{k}$
$l,m,n$ are projections of $AB$ on $X,Y,Z$ respectively.
Projection of $\overrightarrow{AB}$ on $YZ$ plane$=\sqrt{m^2+n^2}=\sqrt{160}$
$m^2+n^2=160\cdots \left(i\right)$
Projection of $\overrightarrow{AB}$ on $ZX$ plane$=\sqrt{l^2+n^2}=\sqrt{153}$
$l^2+n^2=153\cdots \left(ii\right)$
Projection of $\overrightarrow{AB}$ on $XY$ plane$=\sqrt{l^2+m^2}=5$
$l^2+m^2=25\cdots \left(iii\right)$
Adding $\left(i\right),\left(ii\right),\left(iii\right)$
$2(l^2+m^2+n^2)=338$
$l^2+m^2+n^2=169\cdots \left(iv\right)$
Using $\left(i\right)$ and $\left(iv\right)$
$l^2=9$
$l=3$
Using $\left(ii\right)$ and $\left(iv\right)$
$m^2=16$
$m=4$
Using $\left(iii\right)$ and $\left(iv\right)$
$n^2=144$
$n=12$
So, $l=3,m=4,n=12$
Projection of $AB$ on $z-\text{axis}$ is $12$