Question

Consider the points below P(0, −3, 0), Q(5, 1, −2), R(7, 4, 1) 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 the points P, Q, and R, we can use the cross product of two vectors in the plane.

Step-by-step explanation:

To find a vector orthogonal to the plane through the points P, Q, and R, we can use the cross product of two vectors in the plane. Let's consider the vectors PQ and PR. The cross product of these two vectors will give us a vector orthogonal to the plane.

PQ = Q - P = (5 - 0, 1 - (-3), -2 - 0) = (5, 4, -2)

PR = R - P = (7 - 0, 4 - (-3), 1 - 0) = (7, 7, 1)

Now we can find the cross product of PQ and PR:

PQ x PR = (4*(-2) - (-2)*7, (-2)*7 - 5*1, 5*7 - 4*7) = (-8 - (-14), -14 - 5, 35 - 28) = (6, -19, 7)

Therefore, the vector (6, -19, 7) is a nonzero vector orthogonal to the plane through the points P, Q, and R.