Answer to Question #164698 in Geometry for Suleman

Question

Department of Mathematics University of Ghana Math123: Vectors and Geometry Module 4: Exercise 3 Module 4 (Lines in Space)

TK/GAB/JB/EKAS/KD/JDGM 15/02/2021

Consider the line l1 joining the points A(−5, −3, 0) and B(7, 1, 4).

1. Find the parametric equation of the line.

2. Show that the point C(1, −1, 2) lies on the line.

3. Show that the point D(−4, 3, 2) is not on the line.

4. Find the foot of the perpendicular Q from D to the line and determine the shortest distance between the point and the line.

5. Thelinel1 intersectsthelinel2 givenbyr=12i−3j+4k+μ􏰀−8i+3j−k􏰁.

(a) Find the position vector of the point R of intersection.

(b) Find the angle between the two lines.

(c) Find the equations of the internal and external bisectors of the angle between the

 lines l1 and l2.

(d) The point D is on l2. Find a point D′ such that DQ = QD′.

(e) Find the equation of the line l2′ joining R and D′.

(f) Show that l1 is the bisector of angle DRD′. [Line l2′ is the mirror image of the line l2 in l1]

Figure 1:

−→ −−→

 1

Answer 1

Parametric Equation:

If r_0 is the  position vector of the point P_0, then the line must have the form

This is the vector equation of a line in three dimensions. By letting r=<x, y, z>, r_0=<x_0,y_0,z_0>, and v=<a, b, c> we obtain the equation

which leads to the parametric equations of the line passing through the point P_0=(x_0,y_0,z_0) and parallel to the vector v=<a, b, c>: