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) \(\overrightarrow r.[(s-2t)\hat i+(3-t)\hat j+(2s+t)\hat k)=15\)
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) \(\overrightarrow r.[(s-2t)\hat i+(3-t)\hat j+(2s+t)\hat k)=15\)
Solution and Explanation
(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.
(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.
(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.
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Top CBSE CLASS XII Three Dimensional Geometry Questions
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In the following cases, find the distance of each of the given points from the corresponding given plane.
Point Plane
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(d) (-6,0,0) 2x-3y+6z-2=0determine whether the given planes are parallel or perpendicular,and in case they are neither, find the angles between them. (a)7x+5y+6z+30=0 and 3x-y-10z+4=0
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(c)2x-2y+4z+5=0 and 3x-3y+6z-1=0
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Concepts Used:
Plane
A surface comprising all the straight lines that join any two points lying on it is called a plane in geometry. A plane is defined through any of the following uniquely:
- Using three non-collinear points
- Using a point and a line not on that line
- Using two distinct intersecting lines
- Using two separate parallel lines
Properties of a Plane:
- In a three-dimensional space, if there are two different planes than they are either parallel to each other or intersecting in a line.
- A line could be parallel to a plane, intersects the plane at a single point or is existing in the plane.
- If there are two different lines that are perpendicular to the same plane then they must be parallel to each other.
- If there are two separate planes which are perpendicular to the same line then they must be parallel to each other.