Find the area of the triangle formed by the points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9) using the vector product.

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Q: Find the area of the triangle formed by the points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9) using the vector product.

Solution: Area = 0.5 * |AB × AC| = 0, as points are collinear.

Steps: 8

Step 1: Identify the points A, B, and C in 3D space. A(1, 2, 3), B(4, 5, 6), C(7, 8, 9).
Step 2: Calculate the vectors AB and AC. AB = B - A = (4-1, 5-2, 6-3) = (3, 3, 3). AC = C - A = (7-1, 8-2, 9-3) = (6, 6, 6).
Step 3: Use the cross product to find AB × AC. The formula for the cross product in 3D is given by the determinant of a matrix formed by the unit vectors i, j, k and the components of AB and AC.
Step 4: Set up the determinant: |i j k| |3 3 3| |6 6 6|.
Step 5: Calculate the determinant. Since AB and AC are parallel (both are multiples of each other), the cross product will be zero.
Step 6: Find the magnitude of the cross product |AB × AC|, which is 0.
Step 7: Calculate the area of the triangle using the formula Area = 0.5 * |AB × AC|. Since |AB × AC| is 0, the area is 0.
Step 8: Conclude that the area of the triangle is 0 because the points A, B, and C are collinear.