LinAlg4.2.6

From Exampleproblems

Find the vector equation of the plane through the point $A(3,-2,1)\,$ and perpendicular to the vector $4\bar{i}+7\bar{j}-4\bar{k}\,$ .

Given the required plane passes through the point $A(3,-2,1)=3\bar{i}-2\bar{j}+\bar{k}\,$

Also given vector is perpendicular to $4\bar{i}+7\bar{j}-4\bar{k}\,$ .

Therefore the unit vector perpendicular to the plane is $\frac{4\bar{i}+7\bar{j}-4\bar{k}}{\sqrt{16+49+16}}=\frac{1}{9}(4\bar{i}+7\bar{j}-4\bar{k})\,$ .

Therefore,the vector equation of the plane is $(\bar{r}-\bar{a})\cdot\bar{n}=0,\bar{r}\cdot\bar{n}=\bar{a}\cdot\bar{n}\,$

Hence $\bar{r}\cdot\frac{1}{9}(4\bar{i}+7\bar{j}-4\bar{k})=(3\bar{i}-2\bar{j}+\bar{k})\cdot\frac{1}{9}(4\bar{i}+7\bar{j}-4\bar{k})\,$ .

$\bar{r}\cdot(4\bar{i}+7\bar{j}-4\bar{k})=12-14-4=-6\,$ .

$\bar{r}\cdot(-4\bar{i}-7\bar{j}+4\bar{k})=6\,$

Main Page:Linear Algebra

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