If A = 4i + 2j and B = -i + 3j, what is the scalar triple product A · (B × A)? (

If A = 4i + 2j and B = -i + 3j, what is the scalar triple product A · (B × A)? (2023)

If A = 4i + 2j and B = -i + 3j, what is the scalar triple product A · (B × A)? (2023)

Step 1: Identify the vectors A and B. A = 4i + 2j and B = -i + 3j.
Step 2: Write down the formula for the cross product B × A.
Step 3: Set up the determinant for the cross product using the unit vectors i, j, k and the components of B and A.
Step 4: Calculate the determinant: |i j k| |-1 3 0| |4 2 0|.
Step 5: Compute the determinant to find B × A. The result is (0i + 0j + 10k) = 10k.
Step 6: Now, find the dot product A · (B × A). Since B × A = 10k, we need to calculate A · (10k).
Step 7: The dot product A · (10k) is 0 because A has no k component (A = 4i + 2j).
Step 8: Therefore, A · (B × A) = 0.

Vector Operations – The question tests understanding of vector operations, specifically the scalar triple product, which involves the dot product and cross product of vectors.
Properties of the Cross Product – It assesses knowledge of the properties of the cross product, particularly that the cross product of two parallel vectors is zero.
Scalar Triple Product – The scalar triple product A · (B × A) is zero if A and B are coplanar or if A is parallel to B.