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State True or False for the given statement:

The angle between the planes r.\left ( 2\hat{i}-3\hat{j}+\hat{k} \right )=1 and \bar{r}.\left ( \hat{i}-\hat{j} \right )=4 is \cos^{-1}\frac{-5}{\sqrt{58}}


Answers (1)

In vector form, if we take θ as the angle between the two planes

\vec{r}.\vec{n_{1}}=\vec{d_{1}} and \vec{r}.\vec{n_{2}}=\vec{d_{2}}


\theta=\frac{\left | \vec{n_{1}}.\vec{n_{2}} \right |}{\left | \vec{n_{1}} \right |\left | \vec{n_{2}} \right |}

Now, the given planes are \vec{r}.\left ( 2\hat{i}-3\hat{j}+\hat{k} \right )=1 and \vec{r}.\left ( \hat{i}-\hat{j}\right )=4

Here,  \vec{n_{1}}=2 \hat{i}-3\hat{j}+\hat{k} and \vec{n_{2}}=\hat{i}-\hat{j}


\theta =\cos^{-1}\frac{2(1)+3(1)+1(0)}{\sqrt{2^{2}+(-3)^{2}+1^{2}}\sqrt{1^{2}+(-1)^{2}+0^{2}}}


The statement is False.

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