If l, m, n are the direction cosines of a nor | vedclass

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(D) Let the points be $A(0, 1, 2)$, $B(3, 0, 2)$, and $C(4, 5, 0)$.Vectors in the plane are $\vec{AB} = (3-0)\hat{i} + (0-1)\hat{j} + (2-2)\hat{k} = 3

Vedclass · Accepted Answer

$\frac{12}{\sqrt{74}}$

Vedclass · Answer

(D) Let the points be $A(0, 1, 2)$, $B(3, 0, 2)$, and $C(4, 5, 0)$.Vectors in the plane are $\vec{AB} = (3-0)\hat{i} + (0-1)\hat{j} + (2-2)\hat{k} = 3\hat{i} - \hat{j}$ and $\vec{AC} = (4-0)\hat{i} + (5-1)\hat{j} + (0-2)\hat{k} = 4\hat{i} + 4\hat{j} - 2\hat{k}$.The normal vector $\vec{n}$ is $\vec{AB} 	imes \vec{AC} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 3 & -1 & 0 \ 4 & 4 & -2 \end{vmatrix} = \hat{i}(2-0) - \hat{j}(-6-0) + \hat{k}(12 - (-4)) = 2\hat{i} + 6\hat{j} + 16\hat{k}$.Dividing by $2$, we get the normal vector $\vec{n} = \hat{i} + 3\hat{j} + 8\hat{k}$.The magnitude is $|\vec{n}| = \sqrt{1^2 + 3^2 + 8^2} = \sqrt{1 + 9 + 64} = \sqrt{74}$.The direction cosines are $l = \frac{1}{\sqrt{74}}$, $m = \frac{3}{\sqrt{74}}$, $n = \frac{8}{\sqrt{74}}$.Thus, $|l| + |m| + |n| = \frac{1+3+8}{\sqrt{74}} = \frac{12}{\sqrt{74}}$.

If $l, m, n$ are the direction cosines of a normal to the plane passing through the points $(0, 1, 2)$, $(3, 0, 2)$, and $(4, 5, 0)$, then $|l| + |m| + |n| = $

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