Find the scalar triple product of vectors A = (1, 2, 3), B = (4, 5, 6), and C =
Practice Questions
Q1
Find the scalar triple product of vectors A = (1, 2, 3), B = (4, 5, 6), and C = (7, 8, 9).
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Questions & Step-by-Step Solutions
Find the scalar triple product of vectors A = (1, 2, 3), B = (4, 5, 6), and C = (7, 8, 9).
Correct Answer: 0
Step 1: Identify the vectors A, B, and C. Here, A = (1, 2, 3), B = (4, 5, 6), and C = (7, 8, 9).
Step 2: Calculate the cross product of vectors B and C, denoted as B × C.
Step 3: Use the formula for the cross product: B × C = (B2*C3 - B3*C2, B3*C1 - B1*C3, B1*C2 - B2*C1).
Step 4: Substitute the values from vectors B and C into the formula: B × C = (5*9 - 6*8, 6*7 - 4*9, 4*8 - 5*7).
Step 5: Calculate each component: B × C = (45 - 48, 42 - 36, 32 - 35) = (-3, 6, -3).
Step 6: Now, calculate the dot product of vector A with the result from the cross product: A · (B × C).
Step 7: Use the formula for the dot product: A · (B × C) = A1*(B × C)1 + A2*(B × C)2 + A3*(B × C)3.
Step 8: Substitute the values: A · (B × C) = 1*(-3) + 2*6 + 3*(-3).
Step 9: Calculate the result: A · (B × C) = -3 + 12 - 9 = 0.
Step 10: Conclude that the scalar triple product is 0, indicating that the vectors A, B, and C are coplanar.
Scalar Triple Product – The scalar triple product of three vectors A, B, and C is calculated as A · (B × C), which gives a scalar value representing the volume of the parallelepiped formed by the vectors.
Vector Coplanarity – Vectors are coplanar if the scalar triple product is zero, indicating that they lie in the same plane.
