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# The projection of a line segment on $x,y,z$ axes are $12,\:4,\:3$ respectively, then the length of the line segment is ?

$(a)\:\:13\:\:\:\qquad\:\:(b)\:\:19\:\:\:\qquad\:\:(c)\:\:11\:\:\:\qquad\:\:(d)\:\:None\:of\:these.$

Toolbox:
• Projection of $\overrightarrow a$ on $\overrightarrow b$ is given by $\large\frac{\overrightarrow a.\overrightarrow b}{|\overrightarrow b|}$
Let the line segment be represented in vector form as $\overrightarrow a=a_1\hat i+a_2\hat j+a_3\hat k$
Vector along coordinate axes are $\overrightarrow i,\:\overrightarrow j\:,\overrightarrow k$
Given that projection of $\overrightarrow a$ on $x\:axis$ is 12, that with $y\:axis$ is 4 and
that with $z\:axis$ is 3.
$\Rightarrow\:\overrightarrow a.\hat i=12$ $\Rightarrow\:a_1=12.,$
$\overrightarrow a.\hat j=4$ $\Rightarrow\:a_2=4.$ and $\overrightarrow a.\hat k=3$ $\Rightarrow\:a_3=3$
$\therefore \overrightarrow a=12\hat i+4\hat j+3\hat k$
The length of the line segment = $|\overrightarrow a|=\sqrt {144+16+9}=13$