Question

Find a nonzero vector orthogonal to the plane through the point P(0, 0, -3), Q(4,2, 0), and R(3, 3, 1).

Answer 1

To find the nonzero vector orthogonal to the given plane, we can use the cross product of two vectors in the plane. By subtracting the coordinates of the given points, we find PQ = (4, 2, 3) and PR = (3, 3, 4). Taking the cross product of PQ and PR gives the vector (0, 3, 0).

Step-by-step explanation:

To find a nonzero vector orthogonal to a plane, we can use the cross product of two vectors in the plane. Let's take the vectors PQ and PR. PQ can be found by subtracting the coordinates of Q from P: PQ = Q - P = (4, 2, 0) - (0, 0, -3) = (4, 2, 3). Similarly, PR can be found by subtracting the coordinates of R from P: PR = R - P = (3, 3, 1) - (0, 0, -3) = (3, 3, 4). Now, we can take the cross product of PQ and PR to find a vector orthogonal to the plane:

PQ x PR = (4, 2, 3) x (3, 3, 4) = (6, 12, -6) - (6, 9, -6) = (0, 3, 0).

So, the vector (0, 3, 0) is a nonzero vector orthogonal to the plane.