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State True or False for the given statement: The angle between the line r=\left ( 5 \hat{i}-\hat{j}-4\hat{k} \right )+\lambda\left ( 2 \hat{i}-\hat{j}+k \right ) and the plane r.\left ( 3 \hat{i}-4\hat{j}-\hat{k} \right )+5=0\, \, \, \, \, \, \sin^{-1}\frac{5}{2\sqrt{91}}   is

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We know, the angle \phi between the plane with normal vector n and the  line with direction vector b is denoted by:

\sin\varphi\frac{\vec{b}.\vec{n}}{\left |\vec{b} \right |.\left |\vec{n} \right |}

Given equation of the line is r=\left ( 5 \hat{i}-\hat{j}-4\hat{k} \right )+\lambda\left ( 2 \hat{i}-\hat{j}+k \right )

Hence, its direction vector will be:


Given equation of the plane is \vec{r}.\left (3\hat{i}-4\hat{j}-\hat{k} \right )+5=0

Hence, its normal vector will be:


Thus, we have:

\sin\varphi =\left | \frac{\left ( 2\hat{i}-\hat{j}+\hat{k} \right )\left ( 3\hat{i}-4\hat{j}-\hat{k} \right )}{\sqrt{2^{2}+(-1)^{2}+1^{2}}\sqrt{3^{2}+(-4)^{2}+(-1)^{2}}} \right |\\ \Rightarrow \sin \varphi=\frac{2(3)-1(-4)+1(-1)}{\sqrt{6}\sqrt{26}}=\frac{9}{\sqrt{156}}=\frac{9}{2\sqrt{39}}

\varphi =\sin^{-1}\frac{9}{2\sqrt{39}}

Therefore, the given statement is False.

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