Answer to Question #293624 in Analytic Geometry for Adam

Question #293624

Find the projection of the line segment joining the points (1, βˆ’1, 6) and (4, 3, 2) on the line (π‘₯βˆ’4)/3 = βˆ’π‘¦ = 𝑧/5 .


1
Expert's answer
2022-02-14T17:08:30-0500

The projection of the segment joining the points P(x1,y1,z1)P(x_1, y_1, z_1) and Q(x2,y2,z2)Q(x_2, y_2, z_2) on a line having direction cosines l,m,nl, m, n is ∣l(x2–x1)+m(y2–y1)+n(z2–z1)∣.|l(x_2 – x_1) + m(y_2 – y1) + n(z_2 – z_1)|.


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

The projection of the given segment is


∣335(4βˆ’1)βˆ’135(3+1)+535(2βˆ’6)∣|\dfrac{3}{\sqrt{35}}(4-1)-\dfrac{1}{\sqrt{35}}(3+1)+\dfrac{5}{\sqrt{35}}(2-6)|

=1535=\dfrac{15}{\sqrt{35}}


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