Find the parametric vector, non-parametric vector and Cartesian form of equations of the plane passing through the three non-collinear points $\left(3,6,-2\right),\left(-1,-2,6\right) \mathrm{and} \left(6,4,-2\right).$

1. Types of vectors:

(i) Collinear vectors:

Two non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are collinear if there exist non-zero scalars $x$ and $y$ such that x a→+y b→=0→.

(ii) Non-colinear vectors:

If a→ and b→ are two non-zero non-collinear vectors, then x a→+y b→=0→ ⇒$x=y=0.$

(iii) Co-planar vectors:

If a→ and b→ are two non-zero vectors, then any vector r→ coplanar with a→ and b→ can be uniquely expressed as r→=x a→+y b→, where $x, y$ are scalars.

Also, $\overrightarrow{r}\mathrm{ }=\left(x\left|\overrightarrow{a}\right|\right)\mathrm{ }\hat{a}\mathrm{ }+\mathrm{ }\left(y\left|\overrightarrow{b}\right|\right)\mathrm{ }\hat{b}$

(iv) Representation of a vector $\overrightarrow{r}$ in terms of three given non-coplanar vectors:

If a→,b→,c→ are three given non-coplanar vectors, then every vector r→ in space can be uniquely expressed as r→=x a→+y b→+z c→ for some scalars $x, y$ and $z$.

2. Scalar product (or dot product) of two vectors:

If $\overrightarrow{a},\mathrm{ }\overrightarrow{b}$ are two vectors inclined at an angle $\theta$, then their scalar product is denoted by $\overrightarrow{a} . \overrightarrow{b}$ and defined as  $\overrightarrow{a}\mathrm{ }. \overrightarrow{b}=\left|\overrightarrow{a}\right| \left|\overrightarrow{b}\right|$ $\cos \theta$.

3. Scalar product (or dot product) of two vectors in the component form:

Given two vectors $\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\overrightarrow{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ , then $\overrightarrow{a}\cdot \overrightarrow{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3$.

4. Cross product of two vectors:

If a→, b→ are two vectors inclined at an angle $\theta$, then their vector product is denoted by  $\overrightarrow{a}\mathrm{ }\times \mathrm{ }\overrightarrow{b}$ and defined as $\overrightarrow{a}\mathrm{ }\times \mathrm{ }\overrightarrow{b}=\left|\overrightarrow{a}\right| \left|\overrightarrow{b}\right|$ $\sin \theta$.

5. Vector product (or cross product) of two vectors in the component form:

Given two vectors $\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\overrightarrow{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ , then $\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|\begin{array}{}\\ \\ \end{array}$

6.  Angle between two non-zero vectors:

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be the two non-zero vectors. Then,

(i) The angle between  $\overrightarrow{a}$ and $\overrightarrow{b}$  is found by the following formula $\theta ={\cos }^{-1}\left(\frac{\overrightarrow{a}\cdot \overrightarrow{b}}{\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|}\right)$

(ii) $\overrightarrow{a}$ and $\overrightarrow{b}$ are said to be parallel if the angle between them is $0$ or $\mathrm{\pi }$.

(iii) $\overrightarrow{a}$ and $\overrightarrow{b}$ are said to be perpendicular if the angle between them is $\frac{\mathrm{\pi }}{2}$ or $\frac{3\mathrm{\pi }}{2}$.

(iv) $\overrightarrow{a}·\overrightarrow{b}=0$ if and only if $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular to each other.

(iv) $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{0}$ if and only if $\overrightarrow{a}$ and $\overrightarrow{b}$ are parallel to each other.

7. Finding area:

(i) Area of a triangle:

Area of $∆ABC$ $=\mathrm{ }\frac{1}{2}\left|\overrightarrow{AB} \times \overrightarrow{AC}\right|=\frac{1}{2}\left|\overrightarrow{BC}\times \overrightarrow{BA}\right|=\frac{1}{2}\left|\overrightarrow{CB}\mathrm{ }\times \mathrm{ }\overrightarrow{CA}\right|$

(ii) Area of a quadrilateral:

Area of a plane convex quadrilateral $ABCD$ is $\frac{1}{2}\left|\overrightarrow{\mathit{A}\mathit{C}}\mathit{ }\mathit{\times }\mathit{ }\overrightarrow{\mathit{B}\mathit{D}}\right|$, where $AC$ and $BD$ are diagonal.

(iii) Area of a triangle when the vertices are given:

If $\overrightarrow{a},\mathrm{ }\overrightarrow{b},\mathrm{ }\overrightarrow{c}$ are the position vectors of the vertices $A, B, C$of $∆ABC,$ then

Area of $∆ABC=\frac{1}{2}\left|\overrightarrow{a}\times \mathrm{ }\overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}\right|$

8. Vectors as the sides of a triangle:

If a→, b→, c→ are the vectors represented by the sides of a triangle taken in order, then $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$. Conversely, if a→, b→, c→ are three non-collinear vectors, such that a→+b→+c→=0→,then they form the sides of a triangle taken in order.

9. Section formula:

(i) If A and B are two points with position vectors $\overrightarrow{a}\mathrm{ } \text{and} \mathrm{ }\overrightarrow{b}$ respectively, then the position vector of a point $C$ dividing $AB$ in the ratio $m:n$ internally and externally are $\frac{m\mathrm{ }\overrightarrow{b}+n\mathrm{ }\overrightarrow{a}}{m+n}$ and $\frac{m\mathrm{ }\overrightarrow{b}-n\mathrm{ }\overrightarrow{a}}{m-n}$ respectively.

(ii) If $A$ and $B$ are two points with position vectors $\overrightarrow{a}\mathrm{ } \text{and} \mathrm{ }\overrightarrow{b}$ respectively and $m, n$ are positive real numbers, then $m \overrightarrow{OA}+n\mathrm{ }\mathrm{ }\overrightarrow{OB}=\left(m+n\right)\mathrm{ }\mathrm{ }\overrightarrow{OC},$ where $C$ is a point on $AB$ dividing it in the ratio $n:m.$

(iii) Centroid of a triangle:

If $S$ is any point in the plane of a triangle $ABC,$ then SA→+SB→+SC→=3 SG→, where $G$ is the centroid of $∆ ABC.$

10. Scalar triple product:

(i) The dot product of the vector $\overrightarrow{a}$ $\times$ $\overrightarrow{b}$ with the vector $\overrightarrow{c}$ is scalar triple product of three vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ and it is written as $\left(\overrightarrow{a}\times \overrightarrow{b}\right)·\overrightarrow{c}$ and denoted by $\left[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}\right].$ It is a scalar quantity

(ii) Properties of scalar triple product:

(a) $\left(\overrightarrow{a}\times \overrightarrow{b}\right)·\overrightarrow{c}=\overrightarrow{a}·\left(\overrightarrow{b}\times \overrightarrow{c}\right)$ 

(b)  $\left[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}\right]=\left[\overrightarrow{b} \overrightarrow{c} \overrightarrow{a}\right]=\left[\overrightarrow{c} \overrightarrow{a} \overrightarrow{b}\right]=-\left[\overrightarrow{b} \overrightarrow{a} \overrightarrow{c}\right]=-\left[\overrightarrow{c} \overrightarrow{b} \overrightarrow{a}\right]=-\left[\overrightarrow{a} \overrightarrow{c} \overrightarrow{b}\right]$

(c) If $\overrightarrow{a}=\left(a_1, a_2, a_3\right)=a_1\hat{i}+a_2\hat{j}+a_3\hat{k} , \overrightarrow{b}=\left(b_1, b_2, b_3\right)=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ and $\overrightarrow{c}=\left(c_1,c_2,c_3\right)=c_1\hat{i}+c_2\hat{j}+c_3\hat{k},$ then $\left[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}\right]=\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|$

(iii) The volume of the Parallelepiped:

(a) with $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ as Coterminous edges = $\left|\left[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}\right]\right|$

(b) with $A,B,C,D$ as vertices of coterminous edges is $\left|\left[\overrightarrow{AB} \overrightarrow{AC} \overrightarrow{AD}\right]\right|$ cubic units.

(iv) The scalar triple product of three non-zero vectors is zero if, and only if the three vectors are coplanar

(v) If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{p},\overrightarrow{q},\overrightarrow{r}$ are any two systems of three vectors, and if $\overrightarrow{p}=x_1\overrightarrow{a}+y_1\overrightarrow{b}+z_1\overrightarrow{c}$, $\overrightarrow{q}=x_2\overrightarrow{a}+y_2\overrightarrow{b}+z_2\overrightarrow{c}$ and $\overrightarrow{r}=x_3\overrightarrow{a}+y_3\overrightarrow{b}+z_3\overrightarrow{c}$, then $\left[\overrightarrow{p},\overrightarrow{q},\overrightarrow{r}\right]$$=\left|\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ x_3& y_3& z_3\end{array}\right|\left[\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\right]$.

11. Condition on coplanarity of three vectors:

Three vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar if, and only if , there exists $r,s,t\in \mathrm{ℝ}$ such that atleast one of them is non-zero and $r\overrightarrow{a}+s\overrightarrow{b}+t\overrightarrow{c}=\overrightarrow{0}$

12. Vector triple product:

(i) For a given set of three vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c},$ the vector $\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)$ is called a vector triple product

(ii) Vector triple product expansion: 

For any three vectors $\overrightarrow{a}\overrightarrow{,b},\overrightarrow{c}$ we have $\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)=\left(\overrightarrow{a}\cdot \overrightarrow{c}\right)\overrightarrow{b}-\left(\overrightarrow{a}\cdot \overrightarrow{b}\right)\overrightarrow{c}$.

(iii) Properties of vector triple product:

(a) $\left({\overrightarrow{a}}_1+{\overrightarrow{a}}_2\right)\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)={\overrightarrow{a}}_1\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)+{\overrightarrow{a}}_2\times \left(\overrightarrow{b}\times \overrightarrow{c}\right),\left(\lambda \overrightarrow{a}\right)\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)=\lambda \left(\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)\right),\lambda \in \mathrm{ℝ}$

(b) $\overrightarrow{a}\times \left(\left({\overrightarrow{b}}_1+{\overrightarrow{b}}_2\right)\times \overrightarrow{c}\right)=\overrightarrow{a}\times \left({\overrightarrow{b}}_1\times \overrightarrow{c}\right)+\overrightarrow{a}\times \left({\overrightarrow{b}}_2\times \overrightarrow{c}\right),\overrightarrow{a}\times \left(\left(\lambda \overrightarrow{b}\right)\times \overrightarrow{c}\right)=\lambda \left(\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)\right),\lambda \in \mathrm{ℝ}$

(c) $\overrightarrow{a}\times \left(\overrightarrow{b}\times \left({\overrightarrow{c}}_1+{\overrightarrow{c}}_2\right)\right)=\overrightarrow{a}\times \left(\overrightarrow{b}\times {\overrightarrow{c}}_1\right)+\overrightarrow{a}\times \left(\overrightarrow{b}\times {\overrightarrow{c}}_2\right),\overrightarrow{a}\times \left(\overrightarrow{b}\times \left(\lambda \overrightarrow{c}\right)\right)=\lambda \left(\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)\right),\lambda \in \mathrm{ℝ}$  

(d) Vector triple product is not associative.

(iv) Jacobi's identity:

For any three vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c},$ we have $\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)+\overrightarrow{b}\times \left(\overrightarrow{c}\times \overrightarrow{a}\right)+\overrightarrow{c}\times \left(\overrightarrow{a}\times \overrightarrow{b}\right)=\overrightarrow{0}$

(v) Lagrange's identity:

For any four vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c},\overrightarrow{d},$ we have $\left(\overrightarrow{a}\times \overrightarrow{b}\right)\cdot \left(\overrightarrow{c}\times \overrightarrow{d}\right)=\left|\begin{array}{cc}\overrightarrow{a}\cdot \overrightarrow{c}& \overrightarrow{a}\cdot \overrightarrow{d}\\ \overrightarrow{b}\cdot \overrightarrow{c}& \overrightarrow{b}\cdot \overrightarrow{d}\end{array}\right|$.

13. Equation of line passing through a point and given its direction:

(i) Parametric form of vector equation: 

The vector equation of a straight line passing through a fixed point with position vector $\overrightarrow{a}$ and parallel to a given vector $\overrightarrow{b}$ is $\overrightarrow{r}=\overrightarrow{a}+t\overrightarrow{b}$, where $t\in \mathrm{ℝ}$.

(ii) Non-parametric form of vector equation:

The vector equation of a straight line passing through a fixed point with position vector $\overrightarrow{a}$ and parallel to a given vector $\overrightarrow{b}$ is $\left(\overrightarrow{r}-\overrightarrow{a}\right)\times \overrightarrow{b}=\overrightarrow{0}$

(iii) Cartesian form:

The cartesian equation of a straight line passing through point $A(x_1,y_1,z_1)$ and parallel to $\overrightarrow{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ is given by $\frac{x-x_1}{b_1}=\frac{y-y_1}{b_2}=\frac{z-z_1}{b_3}$.

14. Equation of a straight line passing through two points:

(i) Parametric form of vector equation:

The parametric form of vector equation of a line passing through two given points whose position vectors are $\overrightarrow{a}$ and $\overrightarrow{b}$ is given by $\left(\overrightarrow{r}-\overrightarrow{a}\right)\times \left(\overrightarrow{b}-\overrightarrow{a}\right)=\overrightarrow{0}$

(ii) Non-parametric form of vector equation:

The non-parametric form of vector equation of a line passing through two given points whose position vectors are $\overrightarrow{a}$ and $\overrightarrow{b}$ is given by $\left(\overrightarrow{r}-\overrightarrow{a}\right)\times \left(\overrightarrow{b}-\overrightarrow{a}\right)=\overrightarrow{0}$

(iii) Cartesian form:

The cartesian form of equation of a line passing through two given points whose position vectors are $\overrightarrow{a}=x_1\hat{i}+y_1\hat{j}+z_1\hat{k}$ and $\overrightarrow{b}=x_2\hat{i}+y_2\hat{j}+z_2\hat{k}$ is given by $\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}$.

15. Angle between two lines:

(i) The acute angle between two given straight lines $\overrightarrow{r}=\overrightarrow{a}+s\overrightarrow{b}$ and $\overrightarrow{r}=\overrightarrow{c}+s\overrightarrow{d}$ is same as that of the angle between $\overrightarrow{b}$ and $\overrightarrow{d}$ which is given by $\theta ={\cos }^{-1}\left(\frac{|\overrightarrow{b}\cdot \overrightarrow{d}|}{\left|\overrightarrow{b}\right|\left|\overrightarrow{d}\right|}\right)$

(ii) The two given lines with direction ratios $b_1,b_2,b_3$ and $d_1,d_2,d_3$ are perpendicular if and only if $b_1{d}_1+b_2{d}_2+b_3{d}_3=0$

(iii) The two given lines with direction ratios $b_1,b_2,b_3$ and $d_1,d_2,d_3$ are parallel if and only if $\frac{b_1}{d_1}=\frac{b_2}{d_2}=\frac{b_3}{d_3}$. 

(iv) If the direction cosines of two given straight lines are $l_1,m_1,n_1$ and $l_2,m_2,n_2$, then the angle between the lines is given by $\cos \theta =|l_1{l}_2+m_1{m}_2+n_1{n}_2|$

16. Point of intersection of two straight lines:

If $\frac{x-x_1}{a_1}=\frac{y-y_1}{a_2}=\frac{z-z_1}{a_3}\text{and}\frac{x-x_2}{b_1}=\frac{y-y_2}{b_2}=\frac{z-z_2}{b_3}$ are the two intersecting lines, then $\left(x_1+s{a}_1,y_1+s{a}_2,z_1+s{a}_3\right)=\left(x_2+t{b}_1,y_2+t{b}_2,z_2+t{b}_3\right)$ for some value of $s$ and $t$.

 17. Distance between two lines: 

(i) Shortest distance between the two parallel lines:

The shortest distance between the two parallel lines $\overrightarrow{r}=\overrightarrow{a}+s\overrightarrow{b}\text{and}\overrightarrow{r}=\overrightarrow{c}+t\overrightarrow{b}$ is given by $d=\frac{\left|\right(\overrightarrow{c}-\overrightarrow{a})\times \overrightarrow{b}|}{\left|\overrightarrow{b}\right|}$, where $\left|\overrightarrow{b}\right|\ne 0$.

(ii) Shortest distance between the two skew lines:

The shortest distance between the two skew lines $\overrightarrow{r}=\overrightarrow{a}+s\overrightarrow{b}\text{and}\overrightarrow{r}=\overrightarrow{c}+t\overrightarrow{d}$ is given by $\delta =\frac{\left|\right(\overrightarrow{c}-\overrightarrow{a})\cdot (\overrightarrow{b}\times \overrightarrow{d}\left)\right|}{|\overrightarrow{b}\times \overrightarrow{d}|}$, where $\left|\overrightarrow{b}\times \overrightarrow{d}\right|\ne 0$

18. Necessary condition for coplanarity of two lines:

If two lines $\frac{x-x_1}{b_1}=\frac{y-y_1}{b_2}=\frac{z-z_1}{b_3}\text{and}\frac{x-x_2}{d_1}=\frac{y-y_2}{d_2}=\frac{z-z_2}{d_3}$ intersect each other (that is, coplanar) then we have $\left|\begin{array}{ccc}{x}_2-x_1& y_2-y_1& z_2-z_1\\ b_1& b_2& b_3\\ d_1& d_2& d_3\end{array}\right|=0$

19. Equation of a plane:

(i) Cartesian equation of a plane in standard form:

In the equation $ax+by+cz+d=0,$ the direction ratios of normal to the plane are proportional to $a, b, c$ and the vector is given by,n→=ai^+bj^+ck^.

(ii) Equation of a plane perpendicular to a vector and passing through a given point:

(a) Vector form:

The vector equation of a plane passing through a point having position vector a→ and normal to n→ is  $\left(\overrightarrow{r}-\overrightarrow{a}\right)\mathrm{ }·\mathrm{ }\overrightarrow{n}=0 \mathrm{o}\mathrm{r}, \overrightarrow{r}\mathrm{ }·\mathrm{ }\overrightarrow{n}=\overrightarrow{a}\mathrm{ }·\mathrm{ }\overrightarrow{n}$.

(b) Cartesian form:

The Cartesian equation of a plane passing through $\left(x_1, y_1, z_1\right)$ and having direction ratios proportional to $a, b, c$ for its normal is

(iii) Equation of a plane when a normal to the plane and the distance of the plane from the origin are given:

(a) Vector form:

The vector equation of a plane having n^ as a unit vector normal to it and at a distance $\text{'}d\text{'}$ from the origin is r→ · n^=d. If $l, m, n$ are direction cosines of the normal to the plane, then its vector equations r→ · (li^+mj^+nk^)=d. This is the vector equation of the normal form of a plane.

(b) Cartesian form:

If $l, m, n$ are the direction cosines of normal to a plane which is at a distance $p$ from the origin, then the Cartesian equation of the plane is $lx+my+nz=p.$

(iv) Intercept form of a plane:

The Cartesian equation of a plane having intercept $a, b$ and $c$ on  $X, Y$ and $Z$ axes respectively is xa+yb+zc=1.

(v) Equation of a plane passing through three points:

(a) Cartesian form: 

The Cartesian equation of a plane passing through points $\left(x_1, y_1, z_1\right), \left(x_2, y_2, z_2\right)$ and $\left(x_3, y_3, z_3\right)$ is   xyz1x1y1z11x2y2z21x3y3z31=0 or, x-x1y-y1z-z1x2-x1y2-y1z2-z1x3-x1y3-y1z3-z1=0

(b) Vector form:

The vector equation of the plane passing through points having position vectors, a→, b→and c→ is $\overrightarrow{r}\mathrm{ }·\mathrm{ }\left(\overrightarrow{a}\times \overrightarrow{b}\right)+\overrightarrow{r}\mathrm{ }·\mathrm{ }\left(\overrightarrow{b}\times \overrightarrow{c}\right)+\overrightarrow{r}\mathrm{ }·\mathrm{ }\left(\overrightarrow{c}\times \overrightarrow{a}\right)=\overrightarrow{a}\mathrm{ }·\mathrm{ }\left(\overrightarrow{b}\times \overrightarrow{c}\right)$

(vi) Equation of a plane passing through a point and parallel to the two given vectors :

The vector equation of a plane passing through a point having position vector a→ and parallel to vectors b→and c→ is r→=a→+m b→+n c→, where $m$ and $n$ are parameters. Or, $\overrightarrow{r}\mathrm{ }·\mathrm{ }\left(\overrightarrow{b}\times \overrightarrow{c}\right)=\overrightarrow{a}\mathrm{ }·\mathrm{ }\left(\overrightarrow{b}\times \overrightarrow{c}\right)$

(vii) Equation of a plane parallel to another plane:

(a) The equation of a plane parallel to the plane $\overrightarrow{r}\mathrm{ }·\mathrm{ }\overrightarrow{n}=d \mathrm{i}\mathrm{s} \overrightarrow{r}\mathrm{ }·\mathrm{ }\overrightarrow{n}=d_1$.

(b) The equation of a plane parallel to the plane $ax+by+cz+d=0$ is $ax+by+cz+\lambda =0$.

(viii) Equation of family of planes:

The equation of the family of planes containing the line $a_1\mathrm{ }x+b_1\mathrm{ }y+c_1\mathrm{ }z+d_1$$= 0 =$ a2 x+b2 y+c2 z+d2 is $\left(a_1\mathrm{ }x+b_1\mathrm{ }y+c_1\mathrm{ }z+d_1\right)+\lambda$ $\left(a_2\mathrm{ }x+b_2\mathrm{ }y+c_2\mathrm{ }z+d_2\right)=0$, where λ is a parameter.

(xi) Equation of a plane bisecting the angle between two planes:

The equations of the planes bisecting the angles between the planes $a_1\mathrm{ }x+b_1\mathrm{ }y+c_1\mathrm{ }z+d_1=0$ and $a_{2}x+b_{2}y+c_{2}z+d_2=0$ is given by a1x+b1y+c1z+d1a2+b2+c2=±a2x+b2y+c2z+d2a22+b22+c22

20. Angle between two planes:

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

(i) If r→ · n→1=d1 and r→ · n→2=d2 are two planes inclined at an angle θ, then $\cos \theta =\frac{\overrightarrow{n_1}\mathrm{ }·\mathrm{ }\overrightarrow{n_2}}{\left|\overrightarrow{n_1}\right|\mathrm{ }\mathrm{ }\left|\overrightarrow{n_2}\right|}$ and, These planes are parallel, if n→1 is parallel to n→2 and perpendicular, if n→1 · n→2=0

(ii) If $a_{1}x+b_{1}y+c_{1}z+d_1=0$ and $a_{2}x+b_{2}y+c_{2}z+d_2=0$ are Cartesian equations of two planes inclined at an angle θ, then

cos θ=a1 a2+b1 b2+c1 c2a12+b12+c12 a22+b22+c22 and, the planes are parallel, if a1a2=b1b2=c1c2 and, perpendicular, if $a_1{a}_2+b_1{b}_2+c_1{c}_2=0$

20. Length of perpendicular from a point on a plane:

The length of the perpendicular from the point $\left(x_1, y_1 z_1\right)$ to the plane $ax+by+cz+d=0$ is ax1+by1+cz1+da2+b2+c2 and the coordinates $\left(\alpha , \beta , \gamma \right)$ of the foot of the perpendicular are given by α-x1a=β-y1b=γ-z1c=-ax1+by1+cz1+da2+b2+c2

13. Distance between parallel planes:

The distance between the parallel planes $ax+by+cz+d_1=0$ and $ax+by+cz+d_2=0$ is given by d1-d2a2+b2+c2.

21. Angle between a line and a plane:

(i) The angle θ between a line x-x1l=y-y1m=z-z1n and a plane $ax + by + cz + d = 0$ is the complement of the angle between the line and normal to the plane and is given by sin $\theta =\frac{al+bm+cn}{\sqrt{a^2+b^2+c^2} \sqrt{l^2+m^2+n^2}}$

(ii) The angle θ between the line r→=a→+λ b→and the plane r→ · n→=d is given by $\sin \theta =\frac{\overrightarrow{b}\mathrm{ }\mathrm{ }·\mathrm{ }\mathrm{ }\overrightarrow{n}}{\left|\overrightarrow{b}\right|\left|\overrightarrow{n}\right|}$

22. Condition for a line to lie in a plane:

(i) The line r→=a→+λ b→ lies in the plane r→ · n→=d, $\mathrm{i}\mathrm{f} \overrightarrow{a}\mathrm{ }·\mathrm{ }\overrightarrow{n}=d \text{and} \overrightarrow{b}\mathrm{ }·\mathrm{ }\overrightarrow{n}=0$.

(ii) The line x-x1l=y-y1m=z-z1n lies in the plane $ax + by + cz + d = 0,$ if $a{x}_1+b{y}_1+c{z}_1+d=0$ and $al+ bm + cn = 0$

21. Condition to check coplanarity of lines:

(i) Two lines x-x1l1=y-y1m1=z-z1n1and,x-x2l2=y-y2m2=z-z2n2 are coplanar, if x2-x1y2-y1z2-z1l1m1n1l2m2n2=0 and the equation of the plane containing them is x-x1y-y1z-z1l1m1n1l2m2n2=0 or, x-x2y-y2z-z2l1m1n1l2m2n2=0

(ii) Two lines $\overrightarrow{r}=\overrightarrow{a_1}+\lambda \overrightarrow{b_1\mathrm{ }\mathrm{ }}\text{and}\mathrm{ }\overrightarrow{r}=\overrightarrow{a_2}+\mu \mathrm{ }\overrightarrow{b_1}$ are coplanar, if $\overrightarrow{a_1}\mathrm{ }·\mathrm{ }\mathrm{ }\left(\overrightarrow{b_1}\times \overrightarrow{\mathrm{ }{b}_2}\right)=\overrightarrow{a_2}\mathrm{ }·\mathrm{ }\left(\overrightarrow{b_1} \times \overrightarrow{b_2}\right)$.