Question

Consider the points below: p(3, 0, 3), q(-2, 1, 5), r(4, 2, 7). (a) Find a nonzero vector orthogonal to the plane through the points p, q, and r.

Answer 1

To find a nonzero vector orthogonal to the plane through points P, Q, and R, we calculate the cross product of vectors PQ and PR, resulting in the orthogonal vector (0, 22, -11).

Step-by-step explanation:

To find a nonzero vector orthogonal to the plane through the points P(3, 0, 3), Q(-2, 1, 5), and R(4, 2, 7), we need to use the cross product of two vectors that lie on the plane. These two vectors can be obtained by subtracting the coordinates of the points: vec{PQ} = Q - P and vec{PR} = R - P.

First, let's find the vectors:
vec{PQ} = (-2 - 3, 1 - 0, 5 - 3) = (-5, 1, 2)
vec{PR} = (4 - 3, 2 - 0, 7 - 3) = (1, 2, 4)

Now we find the cross product of vec{PQ} and vec{PR}:
vec{PQ} x vec{PR} = |i j k|
|-5 1 2|
| 1 2 4|
= (1*4 - 2*2)i - (-5*4 - 2*1)j + (-5*2 - 1*1)k
= (4 - 4)i + (20 + 2)j + (-10 - 1)k
= 0i + 22j -11k

Therefore, the nonzero vector orthogonal to the plane is (0, 22, -11).