Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

Let the given points be $A(2,1,-3)$ and $B(-1,0,2)$.

The direction ratios of the line are $3,2,6$.

The direction cosines of the line are $l=\frac{3}{\sqrt{3^2+2^2+6^2}}=\frac{3}{7}$, $m=\frac{2}{7}$, $n=\frac{6}{7}$.

The projection of the line segment $AB$ on the line is given by $|l(x_2-x_1)+m(y_2-y_1)+n(z_2-z_1)|$.

Substituting the values, we get:

Projection $=\left|\frac{3}{7}(-1-2)+\frac{2}{7}(0-1)+\frac{6}{7}(2-(-3))\right|$

$=\left|\frac{3}{7}(-3)+\frac{2}{7}(-1)+\frac{6}{7}(5)\right|$

$=\left|\frac{-9-2+30}{7}\right|$

$=\frac{19}{7}$ units.

Answer: $\frac{19}{7}$ units