300. Let A = (5, −3, 6), B = (0, 0, 0), and C = (3, 7, 1). Show that ABC is a right angle

To show that triangle ABC is a right angle, we can use the dot product of vectors. The dot product of two vectors is zero if and only if the vectors are orthogonal (i.e., the angle between them is 90 degrees).

First, we need to find the vectors AB and AC.

Vector AB = B - A = (0, 0, 0) - (5, -3, 6) = (-5, 3, -6) Vector AC = C - A = (3, 7, 1) - (5, -3, 6) = (-2, 10, -5)

Then, we calculate the dot product of AB and AC.

AB · AC = (-5)(-2) + 310 + (-6)*(-5) = 10 + 30 + 30 = 70

Since the dot product of AB and AC is not zero, the angle between them is not 90 degrees. Therefore, triangle ABC is not a right triangle.