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Equation of a plane perpendicular to a given vector and passing through a given point - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Equation of a plane perpendicular to a given vector and passing through a given point is considered one of the most asked concept.

  • 25 Questions around this concept.

Solve by difficulty

A plane bisects the line segment joining the points (1, 2, 3) and (−3, 4, 5) at right angles. Then this plane also passes through the point :

Find the foot perpendicular from the point (3,7,4) to the plane 2x+4yz=2

Concepts Covered - 1

Equation of a plane perpendicular to a given vector and passing through a given point

Imagine a pair of orthogonal vectors that share an initial point. Visualize grabbing one of the vectors and twisting it. As you twist, the other vector spins around and sweeps out a plane. Here, we describe that concept mathematically. 

Let n=ai^+bj^+ck^ be a vector and P(x0,y0,z0) be a point. Then the set of all point Q(x,y,z) such that PQ orthogonal to n forms a plane.
We say that n is a normal vector, or perpendicular to the plane. Remember, the dot product of orthogonal vectors is zero. This fact generates the vector equation of a plane:
nPQ=0

If position vector of point P is P and position vector of point Q is q, then
(qp)n=0
( As PQ=qp)

This is the vector equation of the plane.

Cartesian form

Position vector of point P and point Q is p=x0i^+y0j^+z0k^ and q=xi^+yj^+zk^ respectively and vector n is ai^+bj^+ck^
Then,
(qp)n=0((xi^+yj^+zk^)(x0i^+y0j^+z0k^))(ai^+bj^+ck^)=0[(xx0)i^+(yy0)j^+(zz0)k^](ai^+bj^+ck^)=0 i.e. a(xx0)+b(yy0)+c(zz0)=0

Thus, the coefficients of x, y, and z in the cartesian equation of a plane are the direction ratios of the normal to the plane.

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Equation of a plane perpendicular to a given vector and passing through a given point

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Reference Books

Equation of a plane perpendicular to a given vector and passing through a given point

Mathematics for Joint Entrance Examination JEE (Advanced) : Vectors and 3D Geometry

Page No. : 4.28

Line : 6

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