Find the distance between Q = (3, -7, -1) and the line through A = (1, 1, 2) and B = (2, 3, 4). This distance is equal to \(\dfrac{\sqrt{d}}{3}\)]for some integer d. What is d?
+128406 The directional vector through AB is given by
(2 -1, 3-1, 4-2) = ( 1, 2, 2) = s
And
QA = ( 1-3, 1- -7, 2- -1) = (-2, 8, 3)
Take the cross-product of QA x s
i j k i j
-2 8 3 -2 8 = [ 16i + 3j - 4k ] - [ 8k + 6i -4j ] = 10i + 7j -12k = (10, 7, -12)
1 2 2 1 2
Distance = length of cross-product √[10^2 + 7^2 + (-12)^2 ] √293 √293
________________________ = ___________________ = _____ = _____
length of s √ [1^2 + 2^2 + 2^2 ] √9 3
So d = 293
