Question

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are \( -2, 1, -1 \) and \( -3, -4, 1 \).

Hint:

Use cross product to find a vector perpendicular to two lines; direction ratios are the components.

Answer 1

Direction ratios: \( \mathbf{a} = (-2, 1, -1) \), \( \mathbf{b} = (-3, -4, 1) \). A vector perpendicular to both is given by their cross product: \[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -2 & 1 & -\\ -3 & -4 & 1 \end{vmatrix} = \hat{i}(1 \cdot 1 - (-1) \cdot (-4)) - \hat{j}((-2) \cdot 1 - (-1) \cdot (-3)) + \hat{k}((-2) \cdot (-4) - 1 \cdot (-3)). \] \[ = \hat{i}(1 - 4) - \hat{j}(-2 - 3) + \hat{k}(8 + 3) = (-3, 5, 11). \] Direction ratios: \( (-3, 5, 11) \).