First slide

Introduction to 3-D Geometry

Question

If the projections of a line segment of length “ $d$ ” on the coordinate axes are $d_{1}, d_{2}, d_{3}$ respectively then $d_{1}^{2} + d_{2}^{2} + d_{3}^{2} =$

Easy

A

$\frac{1}{2} d^{2}$
B

$d^{2}$
C

$2 d^{2}$
D

$3 d^{2}$

Solution

The projection of $A B$ on $x -$ axis is $d_{1} = | x_{2} - x_{1} |$
The projection of $A B$ on $y -$ axis is $d_{2} = | y_{2} - y_{1} |$
The projection of $A B$ on $z -$ axis is $d_{3} = | z_{2} - z_{1} |$
Hence, ${d_{1}}^{2} + {d_{2}}^{2} + {d_{3}}^{2} = {(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z_{1})}^{2} = d^{2}$

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