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 \(d\)?
We can find the direction vector, AB = s , of the line thusly
( 2 - 1, 3 - 1, 4 - 2) = ( 1, 2, 2) = s
Next....find the vector QA = ( 1 - 3, 1 - - 7, 2 - - 1) = ( -2, 8, 3)
Next.....form the cross product of QA x s =
i j k i j
-2 8 3 - 2 8 =
1 2 2 1 2
[ i * 8 * 2 + j * 3 * 1 + k * -2 * 2] - [ k * 8 * 1 + i * 3 * 2 + j * - 2 * 2 ] =
[ 16i + 3j - 4k] - [ 8k + 6i - 4j] =
10i + 7j - 12 k
The distance =
Length of [QA x s]
_________________ =
Length of [ s ]
√ [ 10^2 + 7^2 + (-12)^2 ] √ 293 √ 293
_______________________ = _______ = ______
√ [ 1^2 + 2^2 + 2^2 ] √9 3
So d = 293