Find the area of the triangle formed by the points A(1, 2, 3), B(4, 5, 6), and C

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Question: 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.

Options:

Correct Answer: 0

Solution:

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

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.

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.

Correct Answer: 0

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.

Vector Product – The vector product (cross product) of two vectors gives a vector perpendicular to the plane formed by the two vectors, and its magnitude represents the area of the parallelogram formed by those vectors.
Collinearity – Three points are collinear if they lie on the same straight line, which results in an area of zero for the triangle formed by them.
Area of a Triangle – The area of a triangle can be calculated using the formula Area = 0.5 * |AB × AC|, where AB and AC are vectors from one vertex to the other two vertices.