# Vectors

• Feb 23rd 2010, 06:10 AM
yobacul
Vectors
The question I have a problem with is this:

1. The lines l1 and l2 have vector equations given by:

r1 = 3i - j + 2k + λ(-2i + j + k)
r2 = -2i + 9j - 2k + μ(5i - 2j + k)

Find the values of λand μ when the vector r1 - r2 is perpendicular to both lines.

2. Find the shortest distance between the two lines
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I am getting the values of λand μ both zero, and I do not think it makes sense, since you wouldn't be able to work out part 2.

Any help?

• Feb 23rd 2010, 09:46 AM
Plato
Quote:

Originally Posted by yobacul
The question I have a problem with is this:

1. The lines l1 and l2 have vector equations given by:

r1 = 3i - j + 2k + λ(-2i + j + k)
r2 = -2i + 9j - 2k + μ(5i - 2j + k)

Find the values of λand μ when the vector r1 - r2 is perpendicular to both lines.

2. Find the shortest distance between the two lines

As stated, this question makes no sense.
Let $\displaystyle P:(3,-1,2),~Q:(-2,9,-2),~M=<-2,1,1>,~\&~N=<5,-2,1>$.
The vector $\displaystyle M\times N$ is perpendicular to both lines.
The distance between the two lines is $\displaystyle \frac{{\left| {\overrightarrow {PQ} \cdot \left( {M \times N} \right)} \right|}}{{\left\| {M \times N} \right\|}}$.