The direction ratios of the normal to the pla | vedclass

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Question

(A) Let the points be $A(0,0,1)$,$B(0,1,2)$,and $C(1,0,3)$.The vectors lying on the plane are $\vec{AB} = (0-0, 1-0, 2-1) = (0,1,1)$ and $\vec{AC} = (

Vedclass · Accepted Answer

$(2,1,-1)$

Vedclass · Answer

(A) Let the points be $A(0,0,1)$,$B(0,1,2)$,and $C(1,0,3)$.The vectors lying on the plane are $\vec{AB} = (0-0, 1-0, 2-1) = (0,1,1)$ and $\vec{AC} = (1-0, 0-0, 3-1) = (1,0,2)$.The normal vector $\vec{n}$ to the plane is given by the cross product $\vec{AB} 	imes \vec{AC}$:$\vec{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 0 & 1 & 1 \ 1 & 0 & 2 \end{vmatrix}$$\vec{n} = \hat{i}(1 	imes 2 - 1 	imes 0) - \hat{j}(0 	imes 2 - 1 	imes 1) + \hat{k}(0 	imes 0 - 1 	imes 1)$$\vec{n} = \hat{i}(2) - \hat{j}(-1) + \hat{k}(-1)$$\vec{n} = 2\hat{i} + \hat{j} - \hat{k}$The direction ratios of the normal are $(2,1,-1)$.

The direction ratios of the normal to the plane passing through $(0,0,1)$,$(0,1,2)$,and $(1,0,3)$ are:

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