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Expert: - 2/9/2009


Question
the line l_1 passes through the point A,whose position vector is i-j-5k, & is parallel to the vector i-j-4k. The line l_2 passes through the point B,whose position vector is 2i-9j-14k, & is parallel to the vector 2i+5j+6k.The point P on l_1 & the point Q on l_2 are such that PQ is perpendicular to both l_1 & l_2.a) find the length of PQ.b) find a vector perpendicular to the plane ¥ which contains PQ & l_2. I dont understand this topic. Can you explain? I'm stuck. Much appreciated

Answer
A is (1,-1,-5),  L1 passes through A, direction (1,-1,-4).
B is (2,-9,-14), L2 passes through B, direction (2,5,6).
P on L1, Q on L2, PQ is perpendicular to L1 and L2.

Let L1 be at (c,d,e).  Let L2 be at (f,g,h).  The direction of the joining line must be (c-f, d-g, e-h).

When you take the cross product of either directional vector with
the vector (c-f, d-g, e-h), you must get 0.

That will lead to 6 equations with 6 unknowns, since you will get 3
by taking the cross product of (c-f, d-g, e-h) with (1,-1,-4) and 3
by taking the cross product of (c-f, d-g, e-h) with (2,5,6).

The length of the vector would be √[(f-c)² + (g-d)² + (h-e)²].

All you've got to do is take the cross products and solve those
equations.  Two vectors a (a1,a2,a3) and b (b1,b2,b3) have a cross
product a × b = (a2b3 − a3b2, a3b1 − a1b3, a1b2 − a2b1).

When the direction of A is croseed with (c-f, d-g, e-h),
a 0 should be gotten in each componet.

When the direction of B is crossed with (c-f, d-g, e-h),
a 0 should be gotten in each componet.

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