Plane
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 \(P\left(x_{0}, y_{0}, z_{0}\right)\) and normal to the vector \((A,B,C)\) is given by \(A\left(x-x_{0}\right)+B\left(y-y_{0}\right)+C\left(z-z_{0}\right)=0\).
Intercept form of a plane
- The intercept form of a plane is given by \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\), 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 \(A\left(x_{1}, y_{1}, z_{1}\right)\), \(B\left(x_{2}, y_{2}, z_{2}\right)\) and \(C\left(x_{3}, y_{3}, z_{3}\right)\) is given by:
Normal form of a plane
- The normal form of a plane is given by
\(x \cos \alpha+y \cos \beta+z \cos \gamma-p=0\), where \(\cos \alpha\), \(cos \beta\) and \(cos \gamma\) 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:
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 \(\dfrac{A_{1}}{A_{2}}=\dfrac{B_{1}}{B_{2}}=\dfrac{C_{1}}{C_{2}}\), 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 \(P\left(x_{1}, y_{1}, z_{1}\right)\) and parallel to two non-collinear vectors \(\mathbf{u}\left(a_{1}, b_{1}, c_{1}\right)\) and \(\mathbf{v}\left(a_{2}, b_{2}, c_{2}\right)\) is given by the equation:
Equation of a plane passing through a point and parallel to a given vector
- The plane passing through \(P_{1}\left(x_{1}, y_{1}, z_{1}\right)\) and \(P_{2}\left(x_{2}, y_{2}, z_{2}\right)\), and parallel to the vector \(\mathbf{u}(a, b, c)\) is given by the equation:
* Distance from a point to a plane*
- The distance \(d\) from the point \(P_{1}\left(x_{1}, y_{1}, z_{1}\right)\) to the plane \(A x+B y+C z+D=0\) is determined by the equation:
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:
where
\[a = \begin{vmatrix} B_1 & C_1 \\ B_2 & C_2 \end{vmatrix},\] \[b = \begin{vmatrix} C_1 & A_1 \\ C_2 & A_2 \end{vmatrix},\] \[c = \begin{vmatrix} A_1 & B_1 \\ A_2 & B_2 \end{vmatrix}, \text{ }and\] \[x_1 = \dfrac{b\begin{vmatrix} D_1 & C_1 \\ D_2 & C_2 \end{vmatrix} - c\begin{vmatrix} D_1 & B_1 \\ D_2 & B_2 \end{vmatrix}}{a^2+b^2+c^2},\] \[y_1 = \dfrac{c\begin{vmatrix} D_1 & A_1 \\ D_2 & A_2 \end{vmatrix} - a\begin{vmatrix} D_1 & C_1 \\ D_2 & C_2 \end{vmatrix}}{a^2+b^2+c^2},\] \[z_1 = \dfrac{a\begin{vmatrix} D_1 & B_1 \\ D_2 & B_2 \end{vmatrix} - b\begin{vmatrix} D_1 & A_1 \\ D_2 & A_2 \end{vmatrix}}{a^2+b^2+c^2}\]PYQs
Plane
1) The plane \(x+2y+3z=12\) cuts the axes of coordinates in \(A\), \(B\), \(C\). Find the equations of the circle circumscribing the triangle \(ABC\).
[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 \(3x-y+3z=8\) and passing through the point \((1,1,1)\).
[2018, 12M]
4) Find the projection of the straight line \(\dfrac{x-1}{2}=\dfrac{y-1}{3}=\dfrac{z+1}{-1}\) on the plane \(x+y+2z=6\).
[2018, 10M]
5) Obtain the equation of the plane passing through the points \((2,3,1)\) and \((4,-5,3)\) parallel to \(x- axis\).
[2015, 6M]
6) Verify if the lines: \(\dfrac{x-a+d}{\alpha-\delta}=\dfrac{y-a}{\alpha}=\dfrac{z-a-d}{\alpha+\delta}\) and \(\dfrac{x-b+c}{\beta-\gamma}=\dfrac{y-b}{\beta}=\dfrac{z-b-c}{\beta+\gamma}\) 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 \(ABCD\) having each diagonal \(AC\) and \(BD\) of length \(2a\), is folded along the diagonal \(AC\) so that the planes \(DAC\) and \(BAC\) are at a right angle. Find the shortest distance between \(AB\) and \(DC\).
[2005, 12M]
10) A plane is drawn through the line \(x+y=1\), \(z=0\) to make an angle \(\sin^{-1}\left(\dfrac{1}{3}\right)\) 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 \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=0\) 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 \(y z\left(\dfrac{b}{c}+\dfrac{c}{b}\right)+z x\left(\dfrac{c}{a}+\dfrac{a}{c}\right)+x y\left(\dfrac{a}{b}+\dfrac{b}{a}\right)=0\).
[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]