Some help on vectors needed?

Question

Which of the following lines are parallel or non parallel, intersecting or non-intersecting. If the lines are non-parallel, how can I find µ and λ for which the lines intersect? What is the position vector of the point of intersection?

r1=i+j+2k+λ (3i-2j+4) and r2=2i-j+3k+µ (-6i+4j-8K)

r3= i-j+3k+ λ (i-j+k) and r4=2i+4j+6k+µ (2i+j+3K)

Answer 1

Which of the following lines are parallel or non parallel, intersecting or non-intersecting. If the lines are non-parallel, how can I find µ and λ for which the lines intersect? What is the position vector of the point of intersection?

Lines are parallel if their directional vectors are non-zero multiples of each other.

r1=i+j+2k+λ(3i-2j+4k) and r2=2i-j+3k+µ(-6i+4j-8K)

-2*(3i-2j+4k) = (-6i+4j-8k)

So r1 and r2 are parallel.

r3= i-j+3k+ λ(i-j+k) and r4=2i+4j+6k+µ(2i+j+3k)

The directional vectors are not non-zero multiples of each other, so the lines are not parallel. Now let's see if they intersect or if they are skew (non-parallel lines that do not intersect). Find λ and µ at the point of intersection.

x: 1 + λ = 2 + 2µ
y: -1 - λ = 4 + µ
z: 3 + λ = 6 + 3µ

x + y: 0 = 6 + 3µ
µ = -2

x: 1 + λ = 2 + 2µ = 2 + 2(-2)
1 + λ = -2
λ = -3

For (λ, µ) = (-3, -2), (x,y,z) = (-2, 2, 0).

The position vector of the point of intersection is
-2i + 2j + 0k.

Answer 2

look at coefficients, take dot products and cross products. compare coefficients. I think i remember doing something like that a long time ago in calc 3