If the vector a = (2, 3, 4) and b = (1, 0, -1), what is the scalar triple product a · (b × a)?

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Q: If the vector a = (2, 3, 4) and b = (1, 0, -1), what is the scalar triple product a · (b × a)?

Solution: The scalar triple product is 0 because a · (b × a) = 0.

Steps: 9

Step 1: Understand the vectors. We have vector a = (2, 3, 4) and vector b = (1, 0, -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 = (0*4 - (-1)*3, 4*1 - 2*(-1), 2*0 - 3*1).
Step 4: Calculate each component: First component = 0 + 3 = 3, Second component = 4 + 2 = 6, Third component = 0 - 3 = -3. So, b × a = (3, 6, -3).
Step 5: Now, calculate the dot product a · (b × a). The dot product formula is: (x1, y1, z1) · (x2, y2, z2) = x1*x2 + y1*y2 + z1*z2.
Step 6: Substitute the values: a · (b × a) = (2*3) + (3*6) + (4*-3).
Step 7: Calculate each term: 2*3 = 6, 3*6 = 18, 4*-3 = -12.
Step 8: Add the results: 6 + 18 - 12 = 12.
Step 9: Since the scalar triple product a · (b × a) is not equal to 0, we conclude that the scalar triple product is 12.