orthogonal to AB, So the distance from P = (2,4,−4) to the line through the points A = (−1,−1,−4) and B = (0,−3,−5) is
2007-11-26 13:43:38 · 1 answers · asked by tchoxi2000 1 in Science & Mathematics ➔ Engineering
Find the distance from the point P(2, 3, -4) to the line thru the points A(−1,−1,−4) and B(0,−3,−5).
____________
Calculate the directional vector of the line thru A and B.
AB = = <0+1, -3+1, -5+4> = <1, -2, -1>
Calculate the magnitude of AB.
|| AB || = √[1² + (-2)² + (-1)²] = √(1 + 4 + 1) = √6
Calculate vector AP.
AP =
= <2+1, 4+1, -4+4> = <3, 5, 0>
Take the cross product of AB and AP.
AB X AP = <1, -2, -1> X <3, 5, 0> = <5, -3, 11>
Calculate the magnitude of AB X AP.
|| AB X AP || = √[5² + (-3)² + 11²] = √(25 + 9 + 121) = √155
Let h = distance between point P and line AB.
And let θ = the angle between the vectors AB and AP.
The magnitude of the cross product can also be written as:
|| AB X AP || = || AB || || AP || sinθ
But
sinθ = h / || AP ||
So we have:
|| AB X AP || = || AB || || AP || {h / || AP ||}
|| AB X AP || = || AB || h
h = || AB X AP || / || AB || = √155 / √6 = √(155/6)
h ≈ 5.0826502
2007-11-30 11:58:56 · answer #1 · answered by Northstar 7 · 4⤊ 0⤋