Find the vector form of the equation of the plane passing through the point P (1-23) and has normal vector n <3,1, - 1>
Solution:
Given, point P(1,-2,3), Normal vector = <3,1,-1>
a→=(x0,y0,z0)=(1,−2,3),N→=<A,B,C>=<3,1,−1>\overrightarrow a=(x_0,y_0,z_0)=(1,-2,3),\overrightarrow N=<A,B,C>=<3,1,-1>a=(x0,y0,z0)=(1,−2,3),N=<A,B,C>=<3,1,−1>
Vector equation: (r→−a→).N→=0(\overrightarrow r-\overrightarrow a).\overrightarrow N=0(r−a).N=0
⇒(r→−(i^−2j^+3k^)).(3i^+j^−k^)=0\Rightarrow (\overrightarrow r-(\hat i-2\hat j+3\hat k)).(3\hat i+\hat j-\hat k)=0⇒(r−(i^−2j^+3k^)).(3i^+j^−k^)=0
⇒(r→−i^+2j^−3k^).(3i^+j^−k^)=0\Rightarrow (\overrightarrow r-\hat i+2\hat j-3\hat k).(3\hat i+\hat j-\hat k)=0⇒(r−i^+2j^−3k^).(3i^+j^−k^)=0
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