If (\vec{x} = 3\hat{i} - 6\hat{j} - \hat{k}), (\vec{y} = \hat{i} + 4\hat{j} - \hat{k}) and (\vec{z} = 3\hat{i} - 8\hat{j} + 4\hat{k}), then the magnitude of the projection of (\vec{x} \times \vec{y}) on (\vec{z}) is :

Given: (\vec{x} = 3\hat{i} - 6\hat{j} - \hat{k}), (\vec{y} = \hat{i} + 4\hat{j} - \hat{k})
Magnitude of projection of ((\vec{x} \times \vec{y})) is ((\vec{x} \times \vec{y})\cdot \vec{z} / ||\vec{z}||)
(\vec{x} \times \vec{y} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 3 & -6 & -1 \ 1 & 4 & -1 \end{vmatrix} = (6 - (-4))\hat{i} - (-3 - (-1))\hat{j} + (12 - (-6))\hat{k} = 10\hat{i} + 2\hat{j} + 18\hat{k})
((\vec{x} \times \vec{y}) \cdot \vec{z} = (10\hat{i} + 2\hat{j} + 18\hat{k}) \cdot (3\hat{i} - 8\hat{j} + 4\hat{k}) = 30 - 16 + 72 = 86)
( ||\vec{z}|| = \sqrt{3^2 + (-8)^2 + 4^2} = \sqrt{9 + 64 + 16} = \sqrt{89})
Magnitude of projection = (\frac{86}{\sqrt{89}})
Given: (\vec{x} = 3\hat{i} - 6\hat{j} - \hat{k}), (\vec{y} = \hat{i} + 4\hat{j} - \hat{k}), (\vec{z} = 3\hat{i} - 8\hat{j} + 4\hat{k})
(\vec{x} \times \vec{y} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 3 & -6 & -1 \ 1 & 4 & -1 \end{vmatrix} = 10\hat{i} + 2\hat{j} + 18\hat{k})
Magnitude of projection of (\vec{x} \times \vec{y}) on (\vec{z}) is (\frac{(\vec{x} \times \vec{y}) \cdot \vec{z}}{|\vec{z}|})
((\vec{x} \times \vec{y}) \cdot \vec{z} = (10)(3) + (2)(-8) + (18)(4) = 30 - 16 + 72 = 86)
(|\vec{z}| = \sqrt{3^2 + (-8)^2 + 4^2} = \sqrt{89})
Magnitude of projection = \frac{86}{\sqrt{89}} \approx 9.1)
Let's recalculate (\vec{x} \times \vec{y}):
(\vec{x} \times \vec{y} = \begin{vmatrix} i & j & k \ 3 & -6 & -1 \ 1 & 4 & -1 \end{vmatrix} = (6+4)i - (-3+1)j + (12+6)k = 10i + 2j + 18k)
Projection of (\vec{x} \times \vec{y}) on (\vec{z}) is:
(\frac{(10i + 2j + 18k) \cdot (3i - 8j + 4k)}{\sqrt{3^2 + (-8)^2 + 4^2}} = \frac{30 - 16 + 72}{\sqrt{89}} = \frac{86}{\sqrt{89}})
The magnitude is approximately 9.1. There must be a mistake in the provided solution.