1. The general equation of first degree in \displaystyle x,y,z i.e., \displaystyle ax+by+cz+d=0 always represents a plane.

2. In the equation \displaystyle ax+by+cz+d=0, the direction ratios of normal to the plane are proportional to \displaystyle a,b,c.

3. A vector normal to the plane \displaystyle ax+by+cz+d=0 is
\displaystyle \overrightarrow n=a\widehat i+b\widehat j+c\widehat k.

4. If \displaystyle l,m,n are the direction cosines of normal to a plane which is at a distance \displaystyle p from the origin, then the cartesian equation of the plane is
\displaystyle lx+my+nz=p. This is known as the normal form of a plane.

5. The vector equation of a plane passing through a point having position vector \displaystyle \overrightarrow a and normal to \displaystyle \overrightarrow n is

\displaystyle (\overrightarrow r-\overrightarrow a)\cdot\overrightarrow n=0

or

\displaystyle \overrightarrow r\cdot\overrightarrow n=\overrightarrow a\cdot\overrightarrow n.

6. The cartesian equation of a plane passing through \displaystyle (x_{1},y_{1},z_{1}) and having direction ratios proportional to \displaystyle a,b,c for its normal is

\displaystyle a(x-x_{1})+b(y-y_{1})+c(z-z_{1})=0.

7. The vector equation of a plane having \displaystyle \widehat n as a unit vector normal to it and at a distance \displaystyle d from the origin is \displaystyle \overrightarrow r\cdot\widehat n=d.

If \displaystyle l,m,n are direction cosines of the normal to the plane, then its vector equation is \displaystyle \overrightarrow r\cdot(l\widehat i+m\widehat j+n\widehat k)=d. This is the vector equation of the normal form of a plane.

8. The vector equation of a plane passing through points having position vectors \displaystyle \overrightarrow a,\overrightarrow b,\overrightarrow c is \displaystyle \overrightarrow r\cdot(\overrightarrow a\times\overrightarrow b+\overrightarrow b\times\overrightarrow c+\overrightarrow c\times\overrightarrow a)=\overrightarrow a\cdot(\overrightarrow b\times\overrightarrow c).

9. A vector normal to the plane passing through points \displaystyle A(\overrightarrow a),B(\overrightarrow b) and \displaystyle C(\overrightarrow c) is \displaystyle \overrightarrow{AB}\times\overrightarrow{AC} or \displaystyle \overrightarrow{BC}\times\overrightarrow{BA} or \displaystyle \overrightarrow{CB}\times\overrightarrow{CA} i.e., \displaystyle \overrightarrow a\times\overrightarrow b+\overrightarrow b\times\overrightarrow c+\overrightarrow c\times\overrightarrow a.

10. The cartesian equation of a plane intercepting lengths \displaystyle a,b,c with \displaystyle X,Y,Z axes respectively is \displaystyle \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1.

11. The equation of a plane passing through points \displaystyle (x_{1},y_{1},z_{1}), (x_{2},y_{2},z_{2}) and \displaystyle (x_{3},y_{3},z_{3}) is

\displaystyle \begin{vmatrix} x & y & z & 1\\ x_{1} & y_{1} & z_{1} & 1\\ x_{2} & y_{2} & z_{2} & 1\\ x_{3} & y_{3} & z_{3} & 1 \end{vmatrix}=0

or

\displaystyle \begin{vmatrix} x-x_{1} & y-y_{1} & z-z_{1}\\ x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1}\\ x_{3}-x_{1} & y_{3}-y_{1} & z_{3}-z_{1} \end{vmatrix}=0.

12. The angle between two planes is defined as the angle between their normals.

(i) If \displaystyle \overrightarrow r\cdot\overrightarrow n_{1}=d_{1} and \displaystyle \overrightarrow r\cdot\overrightarrow n_{2}=d_{2} are two planes inclined at an angle \displaystyle \theta, then

\displaystyle \cos\theta= \frac{\overrightarrow n_{1}\cdot\overrightarrow n_{2}} {|\overrightarrow n_{1}||\overrightarrow n_{2}|}.

These planes are parallel if \displaystyle \overrightarrow n_{1} is parallel to \displaystyle \overrightarrow n_{2} and perpendicular if \displaystyle \overrightarrow n_{1}\cdot\overrightarrow n_{2}=0

(ii) If \displaystyle a_{1}x+b_{1}y+c_{1}z+d_{1}=0 and \displaystyle a_{2}x+b_{2}y+c_{2}z+d_{2}=0 are cartesian equations of two planes inclined at an angle \displaystyle \theta, then

\displaystyle \cos\theta= \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}}}.

The planes are parallel if

\displaystyle \frac{a_{1}}{a_{2}}= \frac{b_{1}}{b_{2}}= \frac{c_{1}}{c_{2}}

and perpendicular if

\displaystyle a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}=0.

13. The vector equation of a plane passing through a point having position vector \displaystyle \overrightarrow a and parallel to vectors \displaystyle \overrightarrow b and \displaystyle \overrightarrow c is \displaystyle \overrightarrow r=\overrightarrow a+m\overrightarrow b+n\overrightarrow c, where \displaystyle m,n are parameters, or \displaystyle \overrightarrow r\cdot (\overrightarrow b\times\overrightarrow c) =\overrightarrow a\cdot (\overrightarrow b\times\overrightarrow c).

14. The vector equation of the plane passing through points having position vectors \displaystyle \overrightarrow a,\overrightarrow b,\overrightarrow c is \displaystyle \overrightarrow r= (1-m-n)\overrightarrow a+ m\overrightarrow b+ n\overrightarrow c  [Parametric Form]

or

\displaystyle \overrightarrow r\cdot(\overrightarrow a\times\overrightarrow b) +\overrightarrow r\cdot(\overrightarrow b\times\overrightarrow c) +\overrightarrow r\cdot(\overrightarrow c\times\overrightarrow a) =\overrightarrow a\cdot(\overrightarrow b\times\overrightarrow c) [Non-parametric form].

15. The equation of a plane parallel to the plane

(a) \displaystyle \overrightarrow r\cdot\overrightarrow n=d is \displaystyle \overrightarrow r\cdot\overrightarrow n=d_{1}

(b) \displaystyle ax+by+cz+d=0 is \displaystyle ax+by+cz+\lambda=0.

16. The length of the perpendicular from the point \displaystyle (x_{1},y_{1},z_{1}) to the plane \displaystyle ax+by+cz+d=0 is \displaystyle \frac{|ax_{1}+by_{1}+cz_{1}+d|} {\sqrt{a^{2}+b^{2}+c^{2}}}  and the coordinates \displaystyle (\alpha,\beta,\gamma) of the foot of the perpendicular are given by

\displaystyle \frac{\alpha-x_{1}}{a}= \frac{\beta-y_{1}}{b}= \frac{\gamma-z_{1}}{c} =-\frac{ax_{1}+by_{1}+cz_{1}+d} {a^{2}+b^{2}+c^{2}}.

The coordinates \displaystyle (\alpha,\beta,\gamma) of the image of the point \displaystyle (x_{1},y_{1},z_{1}) in the plane \displaystyle ax+by+cz+d=0 are

\displaystyle \frac{\alpha-x_{1}}{a}= \frac{\beta-y_{1}}{b}= \frac{\gamma-z_{1}}{c} =-2\frac{ax_{1}+by_{1}+cz_{1}+d} {a^{2}+b^{2}+c^{2}}.

17. The distance between the parallel planes \displaystyle ax+by+cz+d_{1}=0 and \displaystyle ax+by+cz+d_{2}=0 is given by \displaystyle \frac{|d_{1}-d_{2}|}{\sqrt{a^{2}+b^{2}+c^{2}}}.

18. The equation of the family of planes containing the line \displaystyle a_{1}x+b_{1}y+c_{1}z+d_{1}=0 and \displaystyle a_{2}x+b_{2}y+c_{2}z+d_{2}=0 is \displaystyle (a_{1}x+b_{1}y+c_{1}z+d_{1})+\lambda(a_{2}x+b_{2}y+c_{2}z+d_{2})=0, where \displaystyle \lambda is a parameter.

19. The equations of the planes bisecting the angles between the planes \displaystyle a_{1}x+b_{1}y+c_{1}z+d_{1}=0 and \displaystyle a_{2}x+b_{2}y+c_{2}z+d_{2}=0 are given by

\displaystyle \frac{a_{1}x+b_{1}y+c_{1}z+d_{1}} {\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}} =\pm \frac{a_{2}x+b_{2}y+c_{2}z+d_{2}} {\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}.

20. The angle \displaystyle \theta between a line \displaystyle \frac{x-x_{1}}{l}=\frac{y-y_{1}}{m}=\frac{z-z_{1}}{n} and a plane \displaystyle ax+by+cz+d=0 is the complement of the angle between the line and normal to the plane and is given by

\displaystyle \sin\theta= \frac{al+bm+cn} {\sqrt{a^{2}+b^{2}+c^{2}}\sqrt{l^{2}+m^{2}+n^{2}}}.

The angle \displaystyle \theta between the line \displaystyle \overrightarrow r=\overrightarrow a+\lambda\overrightarrow b and the plane \displaystyle \overrightarrow r\cdot\overrightarrow n=d is given by

\displaystyle \sin\theta= \frac{|\overrightarrow b\cdot\overrightarrow n|} {|\overrightarrow b|,|\overrightarrow n|}.

A line is parallel to a plane if it is perpendicular to the normal to the plane.

A line is perpendicular to a plane if it is parallel to the normal to the plane.

21. The line \displaystyle \overrightarrow r=\overrightarrow a+\lambda\overrightarrow b lies in the plane \displaystyle  \overrightarrow r\cdot\overrightarrow n=d, if \displaystyle \overrightarrow a\cdot\overrightarrow n=d and \displaystyle \overrightarrow b\cdot\overrightarrow n=0.

The line \displaystyle \frac{x-x_{1}}{l}=\frac{y-y_{1}}{m}=\frac{z-z_{1}}{n} lies in the plane \displaystyle ax+by+cz+d=0, if \displaystyle ax_{1}+by_{1}+cz_{1}+d=0 and \displaystyle al+bm+cn=0.

22. The equation of a plane containing the line \displaystyle \frac{x-x_{1}}{l}=\frac{y-y_{1}}{m}=\frac{z-z_{1}}{n} is \displaystyle a(x-x_{1})+b(y-y_{1})+c(z-z_{1})=0, where \displaystyle al+bm+cn=0.

23. Two lines \displaystyle \frac{x-x_{1}}{l_{1}}=\frac{y-y_{1}}{m_{1}}=\frac{z-z_{1}}{n_{1}} and \displaystyle \frac{x-x_{2}}{l_{2}}=\frac{y-y_{2}}{m_{2}}=\frac{z-z_{2}}{n_{2}} are coplanar, if

\displaystyle \begin{vmatrix} x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1}\\ l_{1} & m_{1} & n_{1}\\ l_{2} & m_{2} & n_{2} \end{vmatrix}=0 and the equation of the plane containing them is

\displaystyle \begin{vmatrix} x-x_{1} & y-y_{1} & z-z_{1}\\ l_{1} & m_{1} & n_{1}\\ l_{2} & m_{2} & n_{2} \end{vmatrix}=0

or

\displaystyle \begin{vmatrix} x-x_{2} & y-y_{2} & z-z_{2}\\ l_{1} & m_{1} & n_{1}\\ l_{2} & m_{2} & n_{2} \end{vmatrix}=0.

24. Two lines \displaystyle \overrightarrow r=\overrightarrow a_{1}+\lambda\overrightarrow b_{1} and \displaystyle  \overrightarrow r=\overrightarrow a_{2}+\mu\overrightarrow b_{2} are coplanar, if \displaystyle \overrightarrow a_{1}\cdot(\overrightarrow b_{1}\times\overrightarrow b_{2}) \overrightarrow a_{2}\cdot(\overrightarrow b_{1}\times\overrightarrow b_{2}).

Discover more from ICSE / ISC / CBSE Mathematics Portal for K12 Students

Subscribe to get the latest posts sent to your email.