Question

Find the cartesian equations of the following planes:

\(\overrightarrow r.(\hat i+\hat j-\hat k)=2\) (b) \(\overrightarrow r.(2\hat i+3\hat j-4\hat k)=1\)

(c) \(\overrightarrow r.[(s-2t)\hat i+(3-t)\hat j+(2s+t)\hat k)=15\)

Answer 1

(a)It is given that equation of the plane is

\(\overrightarrow r.(\hat i+\hat j-\hat k)=2\)...(1)

For any arbitrary point P(x,y,z) on the plane, position vector r→ is given by,
\(\overrightarrow r.(x\hat i+y\hat j-z\hat k)=z\hat k\)

Substituting the value of \(\overrightarrow r\) in equation(1), we obtain
\((x\hat i+y\hat j-z\hat k).(\hat i+\hat j-\hat k)=2\) 
\(\Rightarrow \) x+y-z=2

This is the cartesian equation of the plane.

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(b) \(\overrightarrow r.(2\hat i+3\hat j-4\hat k)=1\)...(1)

For any arbitrary point P(x,y,z) on the plane, position vector \(\overrightarrow r\) is given by,
\(\overrightarrow r.(x\hat i+y\hat j-z\hat k)\)

Substituting the value of \(\overrightarrow r\) in equation(1), we obtain
\((x\hat i+y\hat j+z\hat k)=z\hat k (2\hat i+3\hat j-4\hat k)=1\)   
\(\Rightarrow\) 2x+3y-4z=1

This is the cartesian equation of the plane.

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(c) \(\overrightarrow r.[(s-2t)\hat i+(3-t)\hat j+(2s+t)\hat k)=15\)...(1)

For any arbitrary point P(x,y,z) on the plane, position vector\(\overrightarrow r\) is given by,
\(\overrightarrow r.(x\hat i+y\hat j-z\hat k)\)

Substituting the value of r→ in equation(1), we obtain
\(\overrightarrow r.(x\hat i+y\hat j-z\hat k)\).\([(s-2t)\hat i+(3-t)\hat j+(2s+t)\hat k)=15\)
\(\Rightarrow \) (s-2t)x+(3-t)y+(2s+t)z=15

This is the cartesian equation of the given plane.