Practice

1. Determine all k1, k2, and k3 such that:

k1(2, -1, 4) + k2(1, 3, -2) + k3(3, 2, 1) = (7, 5, 6)

2. Given three vectors:

u = (3, -2, 1), v = (-1, 5, 0), w = (4, 1, 2)

a). Show whether {(u, v, w} is an orthogonal set

b). Find the length of the orthogonal projection of vector u onto vector w

3. Given 4 points in space:

A(1, 0, 2); B(2, -1, 3); C(0, 2, 1); P(3, 1, 4)

a). Find the equation of the plane passing through points A, B, and C in vector form

b). Find the equation of the plane passing through points A, B, and C using the plane’s normal vector

c). Calculate the distance from point P to that plane