The perimeter of the triangle whose vertices | vedclass

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(B) Let the vertices be $A$,$B$,and $C$ with position vectors $\vec{a} = \hat{i}+\hat{j}+\hat{k}$,$\vec{b} = 5\hat{i}+3\hat{j}-3\hat{k}$,and $\vec{c}

Vedclass · Accepted Answer

$(15+\sqrt{157}) 	ext{ units}$

Vedclass · Answer

(B) Let the vertices be $A$,$B$,and $C$ with position vectors $\vec{a} = \hat{i}+\hat{j}+\hat{k}$,$\vec{b} = 5\hat{i}+3\hat{j}-3\hat{k}$,and $\vec{c} = 2\hat{i}+5\hat{j}+9\hat{k}$.The side lengths are the magnitudes of the vectors $\vec{AB}$,$\vec{BC}$,and $\vec{AC}$.$\vec{AB} = \vec{b} - \vec{a} = (5-1)\hat{i} + (3-1)\hat{j} + (-3-1)\hat{k} = 4\hat{i} + 2\hat{j} - 4\hat{k}$.$|\vec{AB}| = \sqrt{4^2 + 2^2 + (-4)^2} = \sqrt{16 + 4 + 16} = \sqrt{36} = 6$.$\vec{BC} = \vec{c} - \vec{b} = (2-5)\hat{i} + (5-3)\hat{j} + (9-(-3))\hat{k} = -3\hat{i} + 2\hat{j} + 12\hat{k}$.$|\vec{BC}| = \sqrt{(-3)^2 + 2^2 + 12^2} = \sqrt{9 + 4 + 144} = \sqrt{157}$.$\vec{AC} = \vec{c} - \vec{a} = (2-1)\hat{i} + (5-1)\hat{j} + (9-1)\hat{k} = 1\hat{i} + 4\hat{j} + 8\hat{k}$.$|\vec{AC}| = \sqrt{1^2 + 4^2 + 8^2} = \sqrt{1 + 16 + 64} = \sqrt{81} = 9$.The perimeter of the triangle is $|\vec{AB}| + |\vec{BC}| + |\vec{AC}| = 6 + \sqrt{157} + 9 = 15 + \sqrt{157}$ units.

The perimeter of the triangle whose vertices have the position vectors $\hat{i}+\hat{j}+\hat{k}$,$5\hat{i}+3\hat{j}-3\hat{k}$,and $2\hat{i}+5\hat{j}+9\hat{k}$ is

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