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(A) Let the position vectors of the vertices be $\vec{A} = 2\hat{i} + 4\hat{j} - \hat{k}$,$\vec{B} = 4\hat{i} + 5\hat{j} + \hat{k}$,and $\vec{C} = 3\h

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$\angle A$

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(A) Let the position vectors of the vertices be $\vec{A} = 2\hat{i} + 4\hat{j} - \hat{k}$,$\vec{B} = 4\hat{i} + 5\hat{j} + \hat{k}$,and $\vec{C} = 3\hat{i} + 6\hat{j} - 3\hat{k}$.First,find the vectors representing the sides:$\vec{AB} = \vec{B} - \vec{A} = (4-2)\hat{i} + (5-4)\hat{j} + (1-(-1))\hat{k} = 2\hat{i} + \hat{j} + 2\hat{k}$.$\vec{BC} = \vec{C} - \vec{B} = (3-4)\hat{i} + (6-5)\hat{j} + (-3-1)\hat{k} = -\hat{i} + \hat{j} - 4\hat{k}$.$\vec{AC} = \vec{C} - \vec{A} = (3-2)\hat{i} + (6-4)\hat{j} + (-3-(-1))\hat{k} = \hat{i} + 2\hat{j} - 2\hat{k}$.Now,check the dot products to identify the right angle:$\vec{AB} \cdot \vec{BC} = (2)(-1) + (1)(1) + (2)(-4) = -2 + 1 - 8 = -9 
eq 0$.$\vec{BA} \cdot \vec{BC} = -\vec{AB} \cdot \vec{BC} = 9 
eq 0$.$\vec{AB} \cdot \vec{AC} = (2)(1) + (1)(2) + (2)(-2) = 2 + 2 - 4 = 0$.Since $\vec{AB} \cdot \vec{AC} = 0$,the vectors $\vec{AB}$ and $\vec{AC}$ are perpendicular.Therefore,$\angle A = 90^{\circ}$.

If the position vectors of the vertices of $\Delta ABC$ are $2\hat{i} + 4\hat{j} - \hat{k}$,$4\hat{i} + 5\hat{j} + \hat{k}$,and $3\hat{i} + 6\hat{j} - 3\hat{k}$,then which of the following angles is a right angle?

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