When calculating direction cosines for a line perpendicular to two given lines, always start by finding the cross product of the direction ratios of the lines. Once you have the resulting vector, normalize it by dividing each component by the magnitude of the vector. This will give you the direction cosines. Remember that the cross product is a fundamental operation when dealing with perpendicular vectors in 3D space.
To determine the direction cosines of a line orthogonal to two given lines, represented by direction ratios \( \mathbf{a} = (1, -2, -2) \) and \( \mathbf{b} = (0, 2, 1) \), we first compute their cross product \( \mathbf{a} \times \mathbf{b} \):
\(\mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & -2 & -2 \\ 0 & 2 & 1 \end{vmatrix}\)
\(\mathbf{c} = \mathbf{i}((-2)(1) - (-2)(2)) - \mathbf{j}((1)(1) - (-2)(0)) + \mathbf{k}((1)(2) - (-2)(0))\)
\(\mathbf{c} = \mathbf{i}(-2 + 4) - \mathbf{j}(1) + \mathbf{k}(2)\)
\(\mathbf{c} = 2\mathbf{i} - \mathbf{j} + 2\mathbf{k}\)
The direction ratios of the perpendicular line are therefore (2, -1, 2).
Next, we convert these direction ratios to direction cosines by normalizing the vector. The magnitude of the vector is calculated as: \(\sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3\).
The direction cosines are obtained by dividing each component by the magnitude:
\(\frac{2}{3}, -\frac{1}{3}, \frac{2}{3}\)
The correct direction cosines are \( \frac{2}{3}, -\frac{1}{3}, \frac{2}{3} \).