We will cover following topics

Plane

General equation of the plane

The general equation of a plane in the cartesian coordinate system is given by $A x + B y + C z + D = 0$ , where the vector $(A, B, C)$ is the normal vector to the straight line.

Point direction form of a plane

The point direction form of a plane passing through $P (x_{0}, y_{0}, z_{0})$ and normal to the vector $(A, B, C)$ is given by $A (x - x_{0}) + B (y - y_{0}) + C (z - z_{0}) = 0$ .

Intercept form of a plane

The intercept form of a plane is given by $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ , where $a$ , $b$ and $c$ are the intercepts on $x$ , $y$ and $z$ axes respectively.

Equation of a plane passing through three point

The equation of a plane passing through three points $A (x_{1}, y_{1}, z_{1})$ , $B (x_{2}, y_{2}, z_{2})$ and $C (x_{3}, y_{3}, z_{3})$ is given by:

| \begin{matrix} x - x_{3} & y - y_{3} & z - z_{3} \\ x_{1} - x_{3} & y_{1} - y_{3} & z_{1} - z_{3} \\ x_{2} - x_{3} & y_{2} - y_{3} & z_{2} - z_{3} \end{matrix} | = 0

Normal form of a plane

The normal form of a plane is given by

$x \cos α + y \cos β + z \cos γ - p = 0$ , where $\cos α$ , $c o s β$ and $c o s γ$ are the direction cosines of any straight line normal to the plane.

Angle between two planes

The dihedral angle between two planes is given by:

\cos φ = \frac{n_{1} \cdot n_{2}}{| n_{1} | \cdot | n_{2} |} = \frac{A_{1} A_{2} + B_{1} B_{2} + C_{1} C_{2}}{\sqrt{A_{1}^{2} + B_{1}^{2} + C_{1}^{2}} \sqrt{A_{2}^{2} + B_{2}^{2} + C_{2}^{2}}}

Condition to determine if two planes are parallel or perperdicular

Two planes $A_{1} x + B_{1} y + C_{1} z + D_{1} = 0$ and $A_{2} x + B_{2} y + C_{2} z + D_{2} = 0$ are:

(i) Parallel, if $\frac{A_{1}}{A_{2}} = \frac{B_{1}}{B_{2}} = \frac{C_{1}}{C_{2}}$ , and

(ii) Pependicular, if $A_{1} A_{2} + B_{1} B_{2} + C_{1} C_{2} = 0$

Equation of a plane passing through a point and parallel to two non-collinear vectors

The plane passing through $P (x_{1}, y_{1}, z_{1})$ and parallel to two non-collinear vectors $u (a_{1}, b_{1}, c_{1})$ and $v (a_{2}, b_{2}, c_{2})$ is given by the equation:

| \begin{matrix} x - x_{1} & y - y_{1} & z - z_{1} \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{matrix} | = 0

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

The plane passing through $P_{1} (x_{1}, y_{1}, z_{1})$ and $P_{2} (x_{2}, y_{2}, z_{2})$ , and parallel to the vector $u (a, b, c)$ is given by the equation:

| \begin{matrix} x - x_{1} & y - y_{1} & z - z_{1} \\ x_{2} - x_{1} & y_{2} - y_{1} & z_{2} - z_{1} \\ a & b & c \end{matrix} | = 0

* Distance from a point to a plane*

The distance $d$ from the point $P_{1} (x_{1}, y_{1}, z_{1})$ to the plane $A x + B y + C z + D = 0$ is determined by the equation:

d = \frac{| A x_{1} + B y_{1} + C z_{1} + D |}{\sqrt{A^{2} + B^{2} + C^{2}}}

Condition for intersection of two planes

If the two planes given by the equations $A_{1} x + B_{1} y + C_{1} z + D_{1} = 0$ and $A_{2} x + B_{2} y + C_{2} z + D_{2} = 0$ respectively interesect, the intersection line is given by:

\frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c},

where

a = | \begin{matrix} B_{1} & C_{1} \\ B_{2} & C_{2} \end{matrix} |,

b = | \begin{matrix} C_{1} & A_{1} \\ C_{2} & A_{2} \end{matrix} |,

c = | \begin{matrix} A_{1} & B_{1} \\ A_{2} & B_{2} \end{matrix} |, a n d

x_{1} = \frac{b | \begin{matrix} D_{1} & C_{1} \\ D_{2} & C_{2} \end{matrix} | - c | \begin{matrix} D_{1} & B_{1} \\ D_{2} & B_{2} \end{matrix} |}{a^{2} + b^{2} + c^{2}},

y_{1} = \frac{c | \begin{matrix} D_{1} & A_{1} \\ D_{2} & A_{2} \end{matrix} | - a | \begin{matrix} D_{1} & C_{1} \\ D_{2} & C_{2} \end{matrix} |}{a^{2} + b^{2} + c^{2}},

z_{1} = \frac{a | \begin{matrix} D_{1} & B_{1} \\ D_{2} & B_{2} \end{matrix} | - b | \begin{matrix} D_{1} & A_{1} \\ D_{2} & A_{2} \end{matrix} |}{a^{2} + b^{2} + c^{2}}

PYQs

Plane

1) The plane $x + 2 y + 3 z = 12$ cuts the axes of coordinates in $A$ , $B$ , $C$ . Find the equations of the circle circumscribing the triangle $A B C$ .

[2019, 10M]

2) Prove that the plane $z = 0$ cuts the enveloping cone of the sphere $x^{2} + y^{2} + z^{2} = 11$ which has the vertex at $(2, 4, 1)$ in a rectangulat hyperbola.

[2019, 10M]

3) Find the equation of the plane parallel to $3 x - y + 3 z = 8$ and passing through the point $(1, 1, 1)$ .

[2018, 12M]

4) Find the projection of the straight line $\frac{x - 1}{2} = \frac{y - 1}{3} = \frac{z + 1}{- 1}$ on the plane $x + y + 2 z = 6$ .

[2018, 10M]

5) Obtain the equation of the plane passing through the points $(2, 3, 1)$ and $(4, - 5, 3)$ parallel to $x - a x i s$ .

[2015, 6M]

6) Verify if the lines: $\frac{x - a + d}{α - δ} = \frac{y - a}{α} = \frac{z - a - d}{α + δ}$ and $\frac{x - b + c}{β - γ} = \frac{y - b}{β} = \frac{z - b - c}{β + γ}$ are coplanar. If yes, find the equation of the plane in which they lie.

[2015, 7M]

7) Find the equation of the plane which passes through the points $(0, 1, 1)$ and $(2, 0, - 1)$ and is parallel to the line joining the points $(- 1, 1, - 2)$ , $(3, - 2, 4)$ . Find also the distance between the line and the plane.

[2013, 10M]

8) The plane $x - 2 y + 3 z = 0$ is rotated through a right angle about its line of intersection with the plane $2 x + 3 y - 4 z - 5 = 0$ , find the equation of the plane in its new position.

[2008, 12M]

9) A square $A B C D$ having each diagonal $A C$ and $B D$ of length $2 a$ , is folded along the diagonal $A C$ so that the planes $D A C$ and $B A C$ are at a right angle. Find the shortest distance between $A B$ and $D C$ .

[2005, 12M]

10) A plane is drawn through the line $x + y = 1$ , $z = 0$ to make an angle $\sin^{- 1} (\frac{1}{3})$ with plane $x + y + z = 5$ .Show that two such planes can be drawn. Find their equations and the angle between them.

[2005, 15M]

11) A variable plane remains at a constant distance unity from the point $(1, 0, 0)$ and cuts the coordinate axes at $A$ , $B$ , and $C$ . Find the locus of the center of the sphere passing the origin and the points $A$ , $B$ and $C$ .

[2003, 12M]

12) A variable plane $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 0$ is parallel to the plane meets the co-ordinate axes of $A$ , $B$ and $C$ . Show that the circle $A B C$ lies on the conic $y z (\frac{b}{c} + \frac{c}{b}) + z x (\frac{c}{a} + \frac{a}{c}) + x y (\frac{a}{b} + \frac{b}{a}) = 0$ .

[2002, 15M]

13) Consider a rectangular parallelopiped with edges $a$ , $b$ and $c$ . Obtain the shortest distance between one of its diagonals and an edge which does not intersect this diagonal.

[2002, 15M]

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