If vector A = (2, 2, 2) and vector B = (1, 1, 1), what is the scalar triple prod

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Question: If vector A = (2, 2, 2) and vector B = (1, 1, 1), what is the scalar triple product A . (B × A)?

Options:

Correct Answer: 0

Solution:

A . (B × A) = 0, since B × A = 0.

If vector A = (2, 2, 2) and vector B = (1, 1, 1), what is the scalar triple product A . (B × A)?

If vector A = (2, 2, 2) and vector B = (1, 1, 1), what is the scalar triple product A . (B × A)?

Step 1: Identify the vectors A and B. A = (2, 2, 2) and B = (1, 1, 1).
Step 2: Calculate the cross product B × A. The formula for the cross product of two vectors (x1, y1, z1) and (x2, y2, z2) is given by: (y1*z2 - z1*y2, z1*x2 - x1*z2, x1*y2 - y1*x2).
Step 3: Substitute the values from B and A into the formula: B × A = (1*2 - 1*2, 2*2 - 2*1, 2*1 - 1*2).
Step 4: Calculate each component: (2 - 2, 4 - 2, 2 - 2) = (0, 2, 0).
Step 5: So, B × A = (0, 0, 0).
Step 6: Now, calculate the dot product A . (B × A). The dot product of two vectors (x1, y1, z1) and (x2, y2, z2) is given by: x1*x2 + y1*y2 + z1*z2.
Step 7: Substitute the values: A . (B × A) = (2*0 + 2*0 + 2*0).
Step 8: Calculate the result: 0 + 0 + 0 = 0.
Step 9: Therefore, the scalar triple product A . (B × A) = 0.

Vector Operations – Understanding the scalar triple product, which involves the dot product and cross product of vectors.
Cross Product Properties – Recognizing that the cross product of two parallel vectors is zero.
Dot Product – Calculating the dot product of a vector with the zero vector results in zero.