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.