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

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

Options:

Correct Answer: 0

Solution:

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

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

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

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.

Scalar Triple Product – The scalar triple product of three vectors a, b, and c is given by a · (b × c), which represents the volume of the parallelepiped formed by the vectors. It is zero if any two vectors are parallel or if one of the vectors is the zero vector.
Cross Product – The cross product of two vectors results in a vector that is perpendicular to the plane formed by the original vectors. The magnitude of the cross product is related to the sine of the angle between the vectors.
Dot Product – The dot product of two vectors results in a scalar and is calculated as the sum of the products of their corresponding components. It is zero if the vectors are orthogonal.