Final answer:
To find a vector perpendicular to the plane through points A, B, and C, calculate the cross product of vectors AB and AC, which are derived from the coordinates of these points. The resulting vector V = (-36, 0, 0) is perpendicular to the plane.
Step-by-step explanation:
To find a vector V that is perpendicular to the plane through the points A=(-4, -4, 1), B=(0, 5, -1), and C=(-2, 5, 3), we need to first calculate two vectors that lie on the plane. We can get these by subtracting the coordinates of the points:A to B and A to C. This gives us:
AB = B - A = (0 + 4, 5 + 4, -1 - 1) = (4, 9, -2)
AC = C - A = (-2 + 4, 5 + 4, 3 - 1) = (2, 9, 2)
Next, to find a vector perpendicular to the plane, we take the cross product of AB and AC. The cross product will give us a vector that is perpendicular to both AB and AC and, by extension, to the plane containing them.
Using the cross product formula, we get:
V = AB x AC = (9*-2 - 2*9, 2*4 - 4*2, 4*9 - 9*4)
So, V = (-18 - 18, 8 - 8, 36 - 36)
V = (-36, 0, 0)
Therefore, the vector V = (-36, 0, 0) is perpendicular to the plane formed by points A, B, and C.