Question

Let \( \mathbf{a} = 4i - 2j + 6k \) and \( \mathbf{b} = 7i + j - 12k \). If \( \mathbf{a} \times \mathbf{b} = \alpha i + \beta j + \gamma k \), then the value of \( \alpha + \beta + \gamma \) equals .............

Hint:

When calculating a cross product, use the determinant of a 3x3 matrix where the first row is the unit vectors and the second and third rows are the components of the two vectors.

Answer 1

Correct Answer: 126

Step 1: Cross product of vectors.
The cross product \( \mathbf{a} \times \mathbf{b} \) is calculated as: \[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 4 & -2 & 6 \\ 7 & 1 & -12 \end{vmatrix} \] Step 2: Determining the components.
Expanding the determinant: \[ \mathbf{a} \times \mathbf{b} = \mathbf{i} \left( \begin{vmatrix} -2 & 6 \\ 1 & -12 \end{vmatrix} \right) - \mathbf{j} \left( \begin{vmatrix} 4 & 6 \\ 7 & -12 \end{vmatrix} \right) + \mathbf{k} \left( \begin{vmatrix} 4 & -2 \\ 7 & 1 \end{vmatrix} \right) \] \[ = \mathbf{i}((-2)(-12) - (6)(1)) - \mathbf{j}((4)(-12) - (6)(7)) + \mathbf{k}((4)(1) - (-2)(7)) \] \[ = \mathbf{i}(24 - 6) - \mathbf{j}(-48 - 42) + \mathbf{k}(4 + 14) \] \[ = 18 \mathbf{i} + 90 \mathbf{j} + 18 \mathbf{k} \] Step 3: Conclusion.
Thus, \( \alpha = 18, \beta = 90, \gamma = 18 \), and the sum \( \alpha + \beta + \gamma = 18 + 90 + 18 = 126 \). The value of \( \alpha + \beta + \gamma \) is 126.