Find the vector equation of a plane passing throught a point with position 2 i - j + k and perpendicular to the vector 4 i +2 j -3 k

We know that the vector equation of the plane passing through a point a and normal to n is, r n = a n Substituting a =2 i - j + k and n =4 i +2 j -3 k , we get r (4 i +2 j -3 k )= (2 i - j + k ) (4 i +2 j -3 k ) r (4 i +2 j -3 k )=8-2-3 r (4 i +2 j -3 k )=3

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Question

Find the vector equation of a plane passing throught a point with position $2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$ and perpendicular to the vector $4\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}$

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Answer

We know that the vector equation of the plane passing through a point $\vec{\text{a}}$ and normal to $\vec{\text{n}}$ is,
$\vec{\text{r}}\cdot\vec{\text{n}}=\vec{\text{a}}\cdot\vec{\text{n}}$
Substituting $\vec{\text{a}}=2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{n}}=4\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}},$ we get
$\vec{\text{r}}\cdot\big(4\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\big)=\big(2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)\cdot\big(4\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\big)$
$\Rightarrow\vec{\text{r}}\cdot\big(4\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\big)=8-2-3$
$\Rightarrow\vec{\text{r}}\cdot\big(4\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\big)=3$

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