Distance Formula :

$\quad$ Given two points $ \ P_1(x_1, y_1) \ $ and $ \ P_2(x_2, y_2) \ $ the distance ( d ) between these points is given by:

$$ d=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}} $$

Distance of $P$ from coordinate axes :

PYQ-2023-3D-Q15

$$P A=\sqrt{y^{2}+z^{2}}, P B=\sqrt{z^{2}+x^{2}}, P C=\sqrt{x^{2}+y^{2}}$$

Section Formula :

PYQ-2023-3D-Q15, PYQ-2023-3D-Q34

• Internal section formula

$$P(x, y)=\left(\frac{m x_2+n x_1}{m+n}, \frac{m y_2+n y_1}{m+n}\right)$$

• External section formula

$$P(x, y)=\left(\frac{m x_2-n x_1}{m-n}, \frac{m y_2-n y_1}{m-n}\right)$$

Direction Cosines And Direction Ratios

PYQ-2023-3D-Q4, PYQ-2023-3D-Q7, PYQ-2023-3D-Q10, PYQ-2023-3D-Q14, PYQ-2023-3D-Q18, PYQ-2023-3D-Q20, PYQ-2023-3D-Q22, PYQ-2023-3D-Q27, PYQ-2023-3D-Q34

• Direction cosines

  Let $\alpha, \beta, \gamma$ be the angles which a directed line makes with the positive directions of the axes of $x, y$ and $z$ respectively,

  then $\cos \alpha, \cos \beta, \cos \gamma$ are called the direction cosines of the line. The direction cosines are usually denoted by $(\ell, \mathrm{m}, \mathrm{n})$.

  Thus $\ell=\cos \alpha, \mathrm{m}=$ $\cos \beta, \mathrm{n}=\cos \gamma$.

• If $\ell, \mathrm{m}, \mathrm{n}$ be the direction cosines of a line, then $\ell^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}=1$

• Direction ratios

  Let $\mathrm{a}, \mathrm{b}, \mathrm{c}$ be proportional to the direction cosines $\ell, \mathrm{m}, \mathrm{n}$ then $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are called the direction ratios

• If $\ell, \mathrm{m}, \mathrm{n}$ be the direction cosines and $\mathrm{a}, \mathrm{b}, \mathrm{c}$ be the direction ratios of a vector, then $ \ell= \pm \frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}}, m= \pm \frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}}, n= \pm \frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}} $

• If the coordinates $P$ and $Q$ are $\left(x_{1}, y_{1}, z_{1}\right)$ and $\left(x_{2}, y_{2}, z_{2}\right)$ then the direction ratios of line $P Q$ are, a $=x_{2}-x_{1}, b=y_{2}-y_{1} \hspace{1mm}$ & $\hspace{1mm} c=z_{2}-z_{1}$ and

  the direction cosines of line $P Q$ are $\ell=\frac{x_{2}-x_{1}}{|P Q|}$, m $=\frac{y_2 - y_1}{|PQ|} $ and $ n =\frac{z_2 - z_1} {|PQ|}$

Angle Between Two Line Segments:

PYQ-2023-3D-Q24

$$\cos \theta=\left|\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\right|$$

• The line will be perpendicular if $a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0$.

• The line will be parallel if $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$.

Projection Of A Line Segment On A Line

$\quad$ If $P (x_{1}, y_{1} , z_{1})$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$ then the projection of $P Q$ on a line having direction cosines $\ell, m, n$ is $|\ell ( x_2 - x_1 ) + m ( y_2 - y_1 ) + n ( z_2 - z_1 )|$

Equation Of A Plane :

PYQ-2023-3D-Q1, PYQ-2023-3D-Q9, PYQ-2023-3D-Q15, PYQ-2023-3D-Q18, PYQ-2023-3D-Q20, PYQ-2023-3D-Q22, PYQ-2023-3D-Q23, PYQ-2023-3D-Q28, PYQ-2023-3D-Q29, PYQ-2023-3D-Q35, PYQ-2023-Vector_Algebra-Q18

$\quad$ General form: $a x+b y+c z+d=0$, where $a, b, c$ are not all zero, $a, b, c, d \in R$.

• Normal form $$\ell x+m y+n z=p$$
• Plane through the point $\left(x_{1}, y_{1}, z_{1}\right)$

$$a\left(x-x_{1}\right)+b\left(y-y_{1}\right)+c\left(z-z_{1}\right)=0$$

• Intercept Form

$$\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}+\frac{\mathrm{z}}{\mathrm{c}}=1$$

• Vector form

$$ (\vec{r}-\vec{a}) \cdot \vec{n}=0 \ \text{or} \ \vec{r} \cdot \vec{n}=\vec{a} \cdot \vec{n} $$

• Any plane parallel to the given plane $\mathrm{ax}+\mathrm{by}+\mathrm{cz}+\mathrm{d}=0$ is $\mathrm{ax}+\mathrm{by}+\mathrm{cz}+\lambda=0$

• Distance between $a x+b y+c z+d_{1}=0$ and $a x+b y+c z+d_{2}=0$ is $d=\frac{\left|d_{1}-d_{2}\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$

• Equation of a plane passing through a given point & parallel to the given vectors

• Parametric form $\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}}+\lambda \overrightarrow{\mathrm{b}}+\mu \overrightarrow{\mathrm{c}}$ where, $\lambda$ and $\mu$ are scalars.

• Non-Parametric form $\quad \overrightarrow{\mathrm{r}} \cdot(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=\overrightarrow{\mathrm{a}} \cdot(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}) \quad$

A Plane and A Point

PYQ-2023-3D-Q1, PYQ-2023-3D-Q2, PYQ-2023-3D-Q5, PYQ-2023-3D-Q9, PYQ-2023-3D-Q14, PYQ-2023-3D-Q17, PYQ-2023-3D-Q30, PYQ-2023-3D-Q32, PYQ-2023-3D-Q33

• Distance of the point $\left(x^{\prime}, y^{\prime}, z^{\prime}\right)$ from the plane $a x+b y+c z+d=0$ is given by $\frac{a x^{\prime}+b y^{\prime}+c z^{\prime}+d}{\sqrt{a^{2}+b^{2}+c^{2}}}$.

• Length of the perpendicular from a point ( $\vec{a}$ ) to plane $\vec{r} \cdot \vec{n}=d$ is given by $p=\frac{|\vec{a} \cdot \vec{n}-d|}{|\vec{n}|}$.

• Foot $\left(x^{\prime}, y^{\prime}, z^{\prime}\right)$ of perpendicular drawn from the point $\left(x_{1}, y_{1}, z_{1}\right)$ to the plane $a x+b y+c z+d=0$ is given by

$$\frac{x^{\prime}-x_{1}}{a}=\frac{y^{\prime}-y_{1}}{b}=\frac{z^{\prime}-z_{1}}{c}=-\frac{\left(a x_{1}+b y_{1}+c z_{1}+d\right)}{a^{2}+b^{2}+c^{2}}$$

• To find image of a point w.r.t. a plane:

$\quad$ Let $P\left(x_{1}, y_{1}, z_{1}\right)$ is a given point and $a x+b y+c z+d=0$ is given plane Let $\left(x^{\prime}, y^{\prime}, z^{\prime}\right)$ is the image point. then

$$\frac{x^{\prime}-x_{1}}{a}=\frac{y^{\prime}-y_{1}}{b}=\frac{z^{\prime}-z_{1}}{c}=-2 \frac{\left(a x_{1}+b y_{1}+c z_{1}+d\right)}{a^{2}+b^{2}+c^{2}}$$

Angle Between Two Planes:

PYQ-2023-3D-Q21, PYQ-2023-3D-Q26

$$\cos \theta=\left|\frac{a a^{\prime}+b b^{\prime}+c c^{\prime}}{\sqrt{a^{2}+b^{2}+c^{2}} \sqrt{a^{\prime 2}+b^{\prime 2}+c^{\prime 2}}}\right|$$

• Planes are perpendicular if $ aa’ + bb’ + cc’ = 0 $

• Planes are parallel if $ \frac{a}{a’} = \frac{b}{b’} = \frac{c}{c’} $

• The angle $\theta$ between the planes $\vec{r} \cdot \vec{n}_1 = d_1 $ and $\vec{r} \cdot \vec{n}_2 = d_2 $ is given by, $\cos \theta=\frac{\vec{n}_1 \cdot \vec{n}_2}{|\vec{n}_1 | \cdot|\vec{n}_2|}$

• Planes are perpendicular if $\vec{n}_{1} \cdot \vec{n}_2 = 0 $ & planes are parallel if $\vec{n}_1 = \lambda \hspace{1mm} \vec{n}_2, \lambda $ is a scalar

Angle Bisectors

• The equations of the planes bisecting the angle between two given 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

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

• Bisector of acute/obtuse angle: First make both the constant terms positive. Then

• $a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}>0 \Rightarrow$ origin lies on obtuse angle

• $a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} < 0 \Rightarrow$ origin lies in acute angle

Family of Planes:

PYQ-2023-3D-Q5, PYQ-2023-3D-Q33

• Any plane through the intersection of $a_{1} x + b_{1} y+c_{1} z + d_{1}= 0 \hspace{1mm}$ & $ \hspace{1mm} a_{2} x+b_{2} y+c_{2} z+d_{2}=0$ is

$$a_{1} x+b_{1} y+c_{1} z+d_{1}+\lambda\left(a_{2} x+b_{2} y+c_{2} z+d_{2}\right)=0$$

• The equation of plane passing through the intersection of the planes $\vec{r} . \vec{n}_1 = d_1 \hspace{1mm} $ & $\hspace{1mm}\vec{r} . \vec{n}_2 = d_2 $ is

  $$\vec{r} \cdot (n_1 + \lambda \vec{n}_2) = d_1 + \lambda d_2 $$

  where $\lambda $ is arbitrary scalar

Area Of Triangle:

PYQ-2023-3D-Q13

$\quad$ From two vector ${\overrightarrow{A B}}$ and ${\overrightarrow{A C}}$. Then area is given by ${\frac{1}{2}|\overrightarrow{A B} \times \overrightarrow{A C}|}$

Volume Of A Tetrahedron:

$\quad$ The volume of a tetrahedron with vertices $ A(x_1, y_1, z_1) , B(x_2, y_2, z_2) , C(x_3, y_3, z_3) $, and $ D(x_4, y_4, z_4) $ is given by

$$ V = \frac{1}{6} \left| \begin{array}{cccc} x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \\ x_4 & y_4 & z_4 & 1 \end{array} \right| $$

Centroid Of A Triangle:

$$ G=\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3}\right) $$

Incentre Of Triangle $A B C$ :

$$ \left(\frac{a x_1+b x_2+c x_3}{a+b+c}, \frac{a y_1+b y_2+c y_3}{a+b+c}, \frac{a z_1+b z_2+c z_3}{a+b+c}\right) $$

Centroid Of A Tetrahedron:

$$ \left(\frac{x_1+x_2+x_3+x_4}{4}, \frac{y_1+y_2+y_3+y_4}{4}, \frac{z_1+z_2+z_3+z_4}{4}\right) $$

If $a, b, c$ are the direction ratios of any line $L$ then $ a \hat{i}+b \hat{j}+c \hat{k}$ will be a vector parallel to the line $\mathrm{L}$.

If $\mathrm{l}, \mathrm{m}$, and $\mathrm{n}$ are the direction cosines of any line $\mathrm{L}$, then $ l \hat{i}+m \hat{j}+n \hat{k} $ is a unit vector parallel to the line $\mathrm{L}$.

If $O P=r$, the direction cosines of $OP$ are $l, m, n$ then the coordinates of $P$ are $(lr, mr, nr)$.

If the direction cosines of the line $A B$ are $I, m, n,|A B|=r$ and the coordinates of $A$ is $\left(x_1, y_1, z_1\right)$ then the coordinates of $B$ are given as $\left(x_1+r l, y_1+r m, z_1+r n\right)$.

If the coordinates $P$ and $Q$ are $\left(x_1, y_1, z_1\right)$ and $\left(x_2, y_2, z_2\right)$ then the direction ratios of line PQ are $a=x_2-x_1, b=y_2-y_1$ and $c=z_2-z_1$ and the direction cosines of line $P Q$ are:

$$l=\frac{x_2-x_1}{|P Q|}, m=\frac{y_2-y_1}{|P Q|} n=\frac{z_2-z_1}{|P Q|}$$

Direction Cosines of x, y, z axis:

• Direction cosines of the $x$-axis is $(1,0,0)$.
• Direction cosines of the $y$-axis is $(0,1,0)$.
• Direction cosines of the $z$-axis is $(0,0,1)$.

Equation Of A Plane Parallel To The Axes:

• Plane parallel to $x$-axis is by $+c z+d=0$

• Plane parallel to $y$-axis is $a x+c z+d=0$

• Plane parallel to $z$-axis is $a x+b y+d=0$

Equation of a plane passing through the origin is $ax+by+cz=0$

Transformation of the equation of a plane to the normal form: $a x+b y+c z-d=0$ in normal form is

$$\frac{a x}{ \pm \sqrt{a^2+b^2+c^2}}+\frac{b y}{ \pm \sqrt{a^2+b^2+c^2}}+\frac{c z}{ \pm \sqrt{a^2+b^2+c^2}}=\frac{d}{ \pm \sqrt{a^2+b^2+c^2}}$$

Area Of Triangle:

$\quad$ Let $A\left(x_1, y_1, z_1\right), B\left(x_2, y_2, z_2\right), C\left(x_3, y_3, z_3\right)$ be the vertices of a triangle, then $$ \Delta=\sqrt{\Delta_x^2+\Delta_y^2+\Delta_z^2} $$ $\quad$ where $$ \Delta_x=\frac{1}{2}\left|\begin{array}{lll} y_1 & z_1 & 1 \\ y_2 & z_2 & 1 \\ y_3 & z_3 & 1 \end{array}\right| $$ $$ \Delta_y=\frac{1}{2}\left|\begin{array}{lll} z_1 & x_1 & 1 \\ z_2 & x_2 & 1 \\ z_3 & x_3 & 1 \end{array}\right| $$ $\quad $ and $$ \Delta_z=\frac{1}{2}\left|\begin{array}{lll} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{array}\right| $$

To Find Image Of A Line With Respect To The Line:

$\quad$ Let $L=\frac{x-x_2}{a}=\frac{y-y_2}{b}=\frac{z-z_2}{c}$ be the given line. Let $\left(x^{\prime}, y^{\prime}, z^{\prime}\right)$ be the image of the point $\left(x_1, y_1, z_1\right)$ with respect to the line $L$. Then

• $ a\left(x_1-x^{\prime}\right)+b\left(y_1-y^{\prime}\right)+c\left(z_1-z^{\prime}\right)=0$

• $ \frac{\frac{x_1+x^{\prime}}{2}-x_2}{a}=\frac{\frac{y_1+y^{\prime}}{2}-y_2}{b}=\frac{\frac{z_1+z^{\prime}}{2}-z_2}{c}=\lambda$

$\quad $ From (ii) get the value of $x^{\prime}, y^{\prime}, z^{\prime}$ in terms of $\lambda$ as $x^{\prime}=2 a \lambda+2 x_2-x_{1,} y^{\prime}=2 b \lambda+2 y_2-y_1$ $z^{\prime}=2 c \lambda+2 z_2-z_1$.

$\quad$ Then put the values of $x^{\prime}, y^{\prime}, z^{\prime}$ in (i) to get $\lambda$ and substitute the value of $\lambda$ to get $\left(x^{\prime}, y^{\prime}, z^{\prime}\right)$.

Angle Between A Line And A Plane:

$\quad$ If $\theta$ is the angle between a line $$\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}$$ $\quad$ and the plane $a x+b y+c z+d=0$, then $$\sin \theta=\left|\frac{a l+b m+c n}{\sqrt{a^2+b^2+c^2} \sqrt{l^2+m^2+n^2}}\right|$$

Skew Lines:

PYQ-2023-3D-Q3, PYQ-2023-3D-Q6, PYQ-2023-3D-Q11, PYQ-2023-3D-Q12, PYQ-2023-3D-Q16, PYQ-2023-3D-Q19, PYQ-2023-3D-Q25, PYQ-2023-3D-Q31

• The straight lines which are not parallel and non-coplanar are called skew lines.

• If $\Delta=\left|\begin{array}{ccc}\alpha^{\prime}-\alpha & \beta^{\prime}-\beta & \gamma^{\prime}-\gamma \\ l & m & n \\ l^{\prime} & m^{\prime} & n^{\prime}\end{array}\right| \neq 0$, then the lines are skew.

• Shortest distance $$ S D=\frac{\left|\begin{array}{ccc} \alpha^{\prime}-\alpha & \beta^{\prime}-\beta & \gamma^{\prime}-\gamma \\ l & m & n \\ l^{\prime} & m^{\prime} & n^{\prime} \end{array}\right|}{\sqrt{\sum\left(m n^{\prime}-m^{\prime} n\right)^2}} $$