Question

Find a nonzero vector orthogonal to the plane through the points A( , , ) and B( , , ), and find the area of triangle ABC.

Answer 1

To find a nonzero vector orthogonal to the plane through the points A and B, you can use the cross product of two vectors in the plane. The area of the triangle ABC can be calculated using the magnitude of the cross product divided by 2.

Step-by-step explanation:

To find a nonzero vector orthogonal to the plane through the points A and B, we can use the cross product of two vectors in the plane. Let's call vector AB a and vector AC b. The cross product of vectors a and b will give us a vector perpendicular to the plane. The area of triangle ABC can be calculated using the magnitude of the cross product divided by 2.

Let's assume A = (x1, y1, z1) and B = (x2, y2, z2). To find the cross product of vectors a and b, we can calculate:

a x b = ((y2 - y1)(z2 - z1) - (z2 - z1)(y2 - y1), (z2 - z1)(x2 - x1) - (x2 - x1)(z2 - z1), (x2 - x1)(y2 - y1) - (y2 - y1)(x2 - x1))

The magnitude of the cross product vector will give us the area of triangle ABC.