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Q.
What is the length of the projection of $3 \hat{ i }+4 \hat{ j }+5 \hat{ k }$ on the xy-plane?
Three Dimensional Geometry
Solution:
$x y$ - plane is perpendicular to $z$ - axis.
Let the vector $a =3 i +4 j +5 k$ make angle $\theta$ with $z$ -axis,
then it makes $90-\theta$ with $xy$ -plane.
unit vector along z-axis is $k$.
So, $\cos \theta =\frac{\vec{ a } \cdot \hat{ k }}{|\vec{ a }| \cdot|\hat{ k }|}=\frac{(3 i +4 j +5 k ) \cdot k }{|3 i +4 j +5 k |} $
$=\frac{5}{5 \sqrt{2}}=\frac{1}{\sqrt{2}}$
$ \Rightarrow \theta=\frac{\pi}{4}$
Hence angle with $x y$ - plane $\frac{\pi}{2}-\frac{\pi}{4}=\frac{\pi}{4}$
projection of $\vec{a}$ on $x y$ plane $=|\vec{a}| \cdot \cos \frac{\pi}{4}$
$=5 \sqrt{2} \times \frac{1}{\sqrt{2}}=5$