Part A 1. Determine whether the following two lines intersect, and if so, find t

Question

Part A 1. Determine whether the following two lines intersect, and if so, find the point of intersection. 1 2t 9 t 5 3t 2. Let v (ar, y, z) and Vo (1,1,1). Describe the s of all points (ar, y, z) such that Ilv voll 2. 3. Let r be the line passing through P (zo, yo, zo) and parallel to v (a, b,c), where a, b, and c are all nonzero. Show that a point (z,y, z) lies on the line r if and only if Side Note: These are called the symmetric equations of a line and are a common way of representing a line without using a parameter.

Explanation / Answer

The parametric equations for r1 and r2 are:-

r1= x=1+2t, y=2-t, z= 4-2t

r2= x= 9+t, y= 5+3t, z= -4-t

The direction vectors for r1 and r2 are < 2, -1, -2 > and < 1, 3, -1 >. The lines are not parallel because their vectors are not proportional.

If the lines were to intersect we would have three equations in two unknowns s and t:

1+2t=9+s (1)

2-t=5+3s (2)

4-2t=-4-s (3)

multiply by 2 in eq 2 and add eq 1 and eq 2

5=19+7s

s= -2

put value of s in eq 1

1+2t=9-2

t= 3

by solving last 2 eq we get same value of t and s

so line do not intersect each other.

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