Question

The vectors \(\vec{a} = 4\mathbf{i} - 3\mathbf{j} - \mathbf{k}\) and \(\vec{b} = 3\mathbf{i} + 2\mathbf{j} + \lambda\mathbf{k}\) are perpendicular to each other. Then the value of \(\lambda\) is equal to:

• A. 3
• B. 4
• C. -3
• D. -4
• E. 6

Hint:

Always remember that the dot product of two perpendicular vectors is zero. This is a key property in vector algebra used to determine orthogonality.

Answer 1

Answer: (E)

For two vectors to be perpendicular, their dot product must be zero: \[ \vec{a} \cdot \vec{b} = (4\mathbf{i} - 3\mathbf{j} - \mathbf{k}) \cdot (3\mathbf{i} + 2\mathbf{j} + \lambda\mathbf{k}) = 0 \] Calculate the dot product: \[ = 4 \times 3 + (-3) \times 2 + (-1) \times \lambda = 12 - 6 - \lambda = 0 \] Solve for \(\lambda\): \[ 6 - \lambda = 0 \] \[ \lambda = 6 \] Thus, the value of \(\lambda\) that makes the vectors perpendicular is 6.