can anybody help me with this question please?
Let L be the line passing through the point P=(5, −5, 3) with direction vector →d=[1, −3, −2]T. Find the shortest distance d from the point P0=(−4, −1, 4) to L, and the point Q on L that is closest to P0. Use the square root symbol '√' where needed to give an exact value for your answer.
d?
Q?
To find the shortest distance (d) from point P0 to line L, we can use vector projection. The point Q on the line L that is closest to P0 can be found by finding the orthogonal projection of the vector P0P onto the direction vector →d of L.
1. Find the vector P0P:
P0P = P - P0 = (5, -5, 3) - (-4, -1, 4) = (9, -4, -1)
2. Use the dot product to find the magnitude of the projection vector:
|proj→d (P0P)| = |→d| * cos(theta) = |→d| * (→d ⋅ P0P) / (|→d| * |P0P|)
|→d| = sqrt(1^2 + (-3)^2 + (-2)^2) = sqrt(1 + 9 + 4) = sqrt(14)
|P0P| = sqrt(9^2 + (-4)^2 + (-1)^2) = sqrt(81 + 16 + 1) = sqrt(98)
→d ⋅ P0P = (1)(9) + (-3)(-4) + (-2)(-1) = 9 + 12 + 2 = 23
|proj→d (P0P)| = (sqrt(14) * 23) / sqrt(98)
3. Calculate the distance d:
d = |P0P - proj→d (P0P)|
d = |P0P| - |proj→d (P0P)| = sqrt(98) - (sqrt(14) * 23) / sqrt(98)
4. Calculate the point Q:
Q = P0 + proj→d (P0P)
Q = (-4, -1, 4) + (sqrt(14) * 23 / sqrt(98)) * (1, -3, -2)
Q = (-4, -1, 4) + (23sqrt(14) / sqrt(98)) * (1, -3, -2)
Therefore, the shortest distance (d) is sqrt(98) - (sqrt(14) * 23) / sqrt(98), and the point Q is (-4 + 23sqrt(14) / sqrt(98), -1 - 69sqrt(14) / sqrt(98), 4 - 46sqrt(14) / sqrt(98)).