SHORT NOTES: Three dimensions

Distance formula :
The distance between two points $P\left ( x_1,y_1,z_1 \right )$ and $Q\left ( x_2,y_2,z_2 \right )$ is

$\sqrt{\left ( x_2-x_1 \right )^2+\left ( y_2-y_1 \right )^2+\left ( z_2-z_1 \right )^2}$

Section formula :

Point dividing line joining $P\left ( x_1,y_1,z_1 \right )$ and $Q\left ( x_2,y_2,z_2 \right )$ in ratio $m:n$ is

${M \equiv \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\,\,\,\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\,\,\,\dfrac{{m{z_2} + n{z_1}}}{{m + n}}} \right)}$ internally

${M \equiv \left( {\dfrac{{m{x_2} - n{x_1}}}{{m - n}},\,\,\,\dfrac{{m{y_2} - n{y_1}}}{{m - n}},\,\,\,\dfrac{{m{z_2} - n{z_1}}}{{m - n}}} \right)}$ externally

Direction cosines and direction ratios :

the direction cosines are given by

$l = \cos \alpha ,\,\,\,m = \cos \beta ,\,\,\,\,n = \cos \gamma$

$\Rightarrow\,\,\,\,{{l^2} + {m^2} + {n^2} = 1}$

The direction ratios are simply a set of three real numbers $a$ , $b$ , $c$ proportional to $l$ , $m$ , $n$ , i.e

$\dfrac{l}{a} = \dfrac{m}{b} = \dfrac{n}{c}$

$\Rightarrow\,\,\,,\,\,{l = \pm \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }},\,\,\,m = \pm \dfrac{b}{{\sqrt {{a^2} + {b^2} + {c^2}} }},\,\,\,n = \pm \dfrac{c}{{\sqrt {{a^2} + {b^2} + {c^2}} }}}$

equation of a plane:

The general equation of a plane is $ax+by+cz+d=0$

Equation of a plane in intercept form is $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$

Equation of plane passing through three points :

$\begin{vmatrix} x-x_1 &y-y_1 &z-z_1 \\ x-x_2& y-y_2 &z-z_2 \\ x-x_3 & y-y_3 & z-z_3 \end{vmatrix}=0$

Parametric form:

Vector equation of plane passing through point having position vector $\overrightarrow{a}$ and parallel to $\overrightarrow{b}$ and $\overrightarrow{c}$ is

$\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}+\omega \overrightarrow{c}$

Non Parametric form:

Vector equation of plane passing through point having position vector $\overrightarrow{a}$ and parallel to $\overrightarrow{b}$ and $\overrightarrow{c}$ is

$\left (\overrightarrow{r}-\overrightarrow{a} \right ).\left ( \overrightarrow{b}x \overrightarrow{c} \right )=0$

Distance of a point from a plane :

Vector form:

Point position vector $\overrightarrow{a}$ and plane $\overrightarrow{r}.\overrightarrow{n}=d$

$p=\frac{\left | \overrightarrow{a}.\overrightarrow{n}-d \right |}{\left | \overrightarrow{n} \right |}$

equation of a straight line in space:

Unsymmetrical form of equation of a line:A line can be defined as the intersection of two planes.

${a_1}x + {b_1}y + {c_1}z + {d_1} = 0$

${a_2}x + {b_2}y + {c_2}z + {d_2} = 0$

The above set of equation represents a line

Symmetric form of equation of line with direction cosines l,m,n and passing trough point $A({x_1},{y_1},{z_1}).$ is : $\dfrac{{x - {x_1}}}{l} = \dfrac{{y - {y_1}}}{m} = \dfrac{{z - {z_1}}}{n}$

Two point form:

line passing through $A\left( {{x_1},{y_1},{z_1}} \right)$ and $B\left( {{x_2},{y_2},{z_2}} \right)$ is

$\dfrac{{x - {x_1}}}{{{x_2} - {x_1}}} = \dfrac{{y - {y_1}}}{{{y_2} - {y_1}}} = \dfrac{{z - {z_1}}}{{{z_2} - {z_1}}}$

Note : point $P\left( {x,y,z} \right)$ at a distance $r$ from $A\left( {{x_1},{y_1},{z_1}} \right)$ along the line with direction cosines $l$ , $m$ , $n$ , we have

$\dfrac{{x - {x_1}}}{l} = \dfrac{{y - {y_1}}}{m} = \dfrac{{z - {z_1}}}{n} = r$

Thus, the coordinates of $P$ can be written as

${x = {x_1} + lr,\,\,\,\,\,y = {y_1} + mr,\,\,\,\,\,z = {z_1} + nr}$

ELECTRICAL ENGINEERING
ENGINEERING MATHEMATICS	Electric Circuits and Fields
Signals and Systems	Electrical Machines
Power Systems	Control Systems
Electrical and Electronic Measurements	Power Electronics and Drives
Analog Electronics	Digital Electronics

Three dimensions

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