Show that the following planes are at right angles. r (2 i - j + … - Vidyadip

Question

Show that the following planes are at right angles.
$\vec{\text{r}}\cdot(2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}})=5$ and $\vec{\text{r}}\cdot(-\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}})=3$

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Answer

We know that the planes $\vec{\text{r}}\cdot\vec{\text{n}}_1=\text{d}_1,\vec{\text{ r}}\cdot\vec{\text{n}}_2=\text{d}_2$ are perpendicular to each other only if $\vec{\text{n}}_1\cdot\vec{\text{n}}_2=0$
Here, $\vec{\text{n}}_1=2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}},\vec{\text{n}}_2=-\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$
Now,
$\vec{\text{n}}_1\cdot\vec{\text{n}}_2=\big(2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)\cdot\big(-\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)$
$=-2+1+1=0$
So, the given planes are perpendicular.

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