Equations of Lines — Question 5

Question 5

Find the line through $P=(2,-1,4)$ whose direction is perpendicular to the directions of both lines $$L_1:\vec r=\left\langle 0,1,2\right\rangle+s\left\langle 1,2,-1\right\rangle,\qquad L_2:\vec r=\left\langle 3,0,-1\right\rangle+t\left\langle 2,-1,1\right\rangle.$$ Explain why exactly one such direction exists.

Question 5 – Solution

Strategy A vector perpendicular to both directions is their cross product.

Direction Compute $$\left\langle 1,2,-1\right\rangle\times\left\langle 2,-1,1\right\rangle=\left\langle 1,-3,-5\right\rangle.$$ Therefore the requested line is $$\boxed{\vec r=\left\langle 2,-1,4\right\rangle+u\left\langle 1,-3,-5\right\rangle}.$$

Verification The new direction has dot product $0$ with each original direction.

Uniqueness The original directions are not parallel, so their common perpendicular directions form the one-dimensional span of their nonzero cross product; reversing direction describes the same line.