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  • State True or False for the given statement: The angle between the planes r. ( 2hat{i}-3hat{j}+hat{k} )=1 and bar{r}.( hat{i}-hat{j} )=4 is cos^{-1}frac{-5}{sqrt{58}}

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}}

Then

\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}

Therefore,

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

=\cos^{-1}\frac{5}{\sqrt{2}\sqrt{14}}\\=\cos^{-1}\frac{5}{2\sqrt{7}}

The statement is False.

Posted by

infoexpert24

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