Answer to Question #293624 in Analytic Geometry for Adam

Question

Find the projection of the line segment joining the points (1, −1, 6) and (4, 3, 2) on the line (𝑥−4)/3 = −𝑦 = 𝑧/5 .

Answer 1

The projection of the segment joining the points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ on a line having direction cosines $l, m, n$ is $|l(x_2 – x_1) + m(y_2 – y1) + n(z_2 – z_1)|.$

$$\sqrt{(3)^2+(-1)^2+(5)^2}=\sqrt{35}$$

The projection of the given segment is

$$|\dfrac{3}{\sqrt{35}}(4-1)-\dfrac{1}{\sqrt{35}}(3+1)+\dfrac{5}{\sqrt{35}}(2-6)|$$

$$=\dfrac{15}{\sqrt{35}}$$