Find the angle between the planes whose vector equations are
\(\overrightarrow r.(2\hat i+2\hat j-3\hat k)=5\) and \(\overrightarrow r.(3\hat i-3\hat j+5\hat k)=3\)
Find the angle between the planes whose vector equations are
\(\overrightarrow r.(2\hat i+2\hat j-3\hat k)=5\) and \(\overrightarrow r.(3\hat i-3\hat j+5\hat k)=3\)
Solution and Explanation
The equations of the given planes are \(\overrightarrow r.(2\hat i+2\hat j-3\hat k)=5\) and \(\overrightarrow r.(3\hat i-3\hat j+5\hat k)=3\)
It is known that if \(\overrightarrow n_1\) and \(\overrightarrow n_2\) are normal to the planes, \(\overrightarrow r.\overrightarrow n_1=\overrightarrow d_1\) and \(\overrightarrow r.\overrightarrow n_2=\overrightarrow d_2\)
then the angle between them, Q, is given by,
cos Q= \(\begin{vmatrix}\frac{\overrightarrow n_1.\overrightarrow n_2}{\|\overrightarrow n_1\| \overrightarrow n_2 \|}\end{vmatrix}\)...(1)
Here, \(\overrightarrow n_1=2\hat i+2\hat j-3\hat k\) and \(\overrightarrow n_2=3\hat i-3\hat j+5\hat k\)
∴ \(\overrightarrow n_1.\overrightarrow n_2\)= \((2\hat i+2\hat j-3\hat k)\)\((3\hat i-3\hat j+5\hat k)\)
=2.3+2.(-3)+(-3).5 =-15
|\(\overrightarrow n_1\)|=\(\sqrt{(2)^2+(2)^2+(-3)^2}=\sqrt{17}\)
|\(\overrightarrow n_2\)|=\(\sqrt{(3)^2+(-3)^2+(5)^2}=\sqrt{43}\)
Substituting the value of \(\overrightarrow n_1.\overrightarrow n_2\),|\(\overrightarrow n_1\)|and|\(\overrightarrow n_2\)| in equation(1), we obtain
cos Q=|\(\frac{-15}{\sqrt{17}.\sqrt{43}}\)|
\(\Rightarrow\) cos Q=\(\frac{15}{\sqrt{731}}\)
\(\Rightarrow\) cos Q-1 =\(\bigg(\frac{15}{\sqrt{731}}\bigg)\)
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Concepts Used:
Angle between Two Planes
A plane is demarcated as two-dimensional in nature and the one which has a flat surface that prolongs infinitely far in two dimensions. It is set up by some stack of lines that are kept together.
Angles between two planes refer to the acute angle which is manifest by the standard vectors of the planes. If the standard vectors of two planes are rectangular, then we can say they are perpendicular. This specific portion tells us what a plane is and what is the angle between the two planes, it also shows us how to calculate the angles between the two planes in a Cartesian plane.