What is the scalar triple product of the vectors (1, 2, 3), (4, 5, 6), and (7, 8

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Question: What is the scalar triple product of the vectors (1, 2, 3), (4, 5, 6), and (7, 8, 9)?

Options:

Correct Answer: 0

Solution:

Scalar triple product = (1, 2, 3) · ((4, 5, 6) × (7, 8, 9)) = 0.

What is the scalar triple product of the vectors (1, 2, 3), (4, 5, 6), and (7, 8, 9)?

What is the scalar triple product of the vectors (1, 2, 3), (4, 5, 6), and (7, 8, 9)?

Step 1: Identify the three vectors: A = (1, 2, 3), B = (4, 5, 6), C = (7, 8, 9).
Step 2: Calculate the cross product of vectors B and C, which is B × C.
Step 3: Use the formula for the cross product: B × C = |i j k| |4 5 6| |7 8 9|.
Step 4: Calculate the determinant to find the cross product: B × C = (5*9 - 6*8)i - (4*9 - 6*7)j + (4*8 - 5*7)k.
Step 5: Simplify the calculations: B × C = (45 - 48)i - (36 - 42)j + (32 - 35)k = (-3)i + (6)j + (-3)k.
Step 6: Now we have B × C = (-3, 6, -3).
Step 7: Next, calculate the dot product of vector A with the result from the cross product: A · (B × C).
Step 8: Use the formula for the dot product: A · (B × C) = (1, 2, 3) · (-3, 6, -3).
Step 9: Calculate the dot product: 1*(-3) + 2*6 + 3*(-3) = -3 + 12 - 9.
Step 10: Simplify the result: -3 + 12 - 9 = 0.
Step 11: The scalar triple product is 0.

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