We consider $3-$ mutually perpendicular lines intersecting at a fixed point. Where this fixed point is called origin $(0)$ and the three line is called coordinate axis labeled as $x-$axis, $y-$axis and $z-$axis respectively as shown in figure.
1) Distance between two points:
Let $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ be the two given points. Then the distance between two points $PQ$ is given by,
$PQ$ = $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$ .......... (i)
2) Section Formula:
Let $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ be any points in the $3-$ dimensional space. Suppose a point $R(x, y, z)$ divides the join of $P$ and $Q$ in the ratio $m:n$. Then,
(i) Internal division:
The coordinates of $R(x, y, z)$ are
$R(x, y, z)$ = $\left ( \frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}, \frac{mz_2+nz_1}{m+n} \right )$ .......... (ii)
(ii) External division:
The coordinates of $R(x, y, z)$ are
$R(x, y, z)$ = $\left ( \frac{mx_2-nx_1}{m-n}, \frac{my_2-ny_1}{m-n}, \frac{mz_2-nz_1}{m-n} \right )$ .......... (iii)
3) Projection:
Projection of a line joining two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ on another line with direction cosine $l, m, n$. Then the direction cosines of $PQ$ are
$\frac{x_2-x_1}{r}, \frac{y_2-y_1}{r}, \frac{z_2-z_1}{r}$ .......... (iv)
Where, $PQ = r = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$
If $\theta$ be the angle between $PQ$ and $AB$ then,
$cos\theta = \frac{x_2-x_1}{r}.l + \frac{y_2-y_1}{r}.m + \frac{z_2-z_1}{r}.n $ .......... (v)
Also the Projection of $PQ$ on $AB$ = $MN$ = $PQcos\theta$
$PQcos\theta = (x_2-x_1).l + (y_2-y_1).m + (z_2-z_1).n $ .......... (vi)
4) Direction Cosine of a line $(l, m, n)$:
If a line $AB$ makes an angle $\alpha, \beta, \gamma$ with the direction $x-$axis, $y-$axis and $z-$axis respectively. Then, $cos\alpha$, $cos\beta$ and $cos\gamma$ are called direction cosines (d.c.s) of a line $AB$. It is generally denoted by $l, m, n$.
$l^2 + m^2 + n^2 = 1$ .......... (vii)
[$Cos\alpha,\;cos\beta,\;cos\gamma$ i.e. $l,\;m,\;n$ are the direction cosines of the line $AB$.]
5) Direction Ratio $(a, b, c)$:
If a set of three numbers $a$, $b$, and $c$ proportional to the directions cosines $l$, $m$, and $n$ of a line, they are called its direction ratios. We have,
$\frac{l}{a}$ = $\frac{m}{b}$ = $\frac{n}{c}$ = $\frac{\sqrt{l^2 + m^2 + n^2}}{\sqrt{a^2 + b^2 + c^2}}$ .......... (viii)
Hence,
$l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}$, $m = \frac{b}{\sqrt{a^2 + b^2 + c^2}}$, $n = \frac{c}{\sqrt{a^2 + b^2 + c^2}}$ .......... (ix)
6) Angle between two lines:
From figure, let, $(l_1,\;m_1,\;n_1)$ and $(l_2,\;m_2,\;n_2)$ be the direction cosines of two given lines $AB$ and $CD$ respectively.
Let $OP$ and $OQ$ be the lines through the origin $O$ parallel to the line $AB$ and $CD$ respectively.
To find the angle between two lines with direction cosine $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$,
$Cos\theta$ = $l_1l_2 + m_1m_2 + n_1n_2$ .......... (x)
(i) Condition of perpendicular of two lines:
$l_1l_2 + m_1m_2 + n_1n_2 = cos\;90^0 = 0$ .......... (xi)
(ii) Condition of parallelism of two lines:
$\frac{l_1}{l_2}$ = $\frac{m_1}{m_2}$ = $\frac{n_1}{n_2}$ = $\frac{\sqrt{l_1\,^2 + m_1\,^2 + n_1\,^2}}{\sqrt{l_2\,^2 + m_2\,^2 + n_2\,^2}}$
∴ $l_1 = l_2$; $m_1 = m_2$, $n_1 = n_2$ .......... (xii)
Which is the required condition of parallelism of two straight lines.
» Angle between two straight lines whose direction ratios are given:
Let $a_1, b_1, c_1$ and $a_2, b_2, c_2$ be the direction ratios of two lines whose corresponding direction cosines are $l_1, m_1, n_1$ and $l_2, m_2, n_2$. Then,
$l_1 = \frac{a_1}{\sqrt{a_1\,^2 + b_1\,^2 + c_1\,^2}}$, $m_1 = \frac{b_1}{\sqrt{a_1\,^2 + b_1\,^2 + c_1\,^2}}$, $n_1 = \frac{c_1}{\sqrt{a_1\,^2 + b_1\,^2 + c_1\,^2}}$
And similarly for $l_2, m_2, n_2$.
» If $\theta$ be the angle between the two lines, then
$cos\theta = l_1l_2 + m_1m_2 + n_1n_2$ = $\frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1\,^2 + b_1\,^2 + c_1\,2}\;\sqrt{a_2\,^2 + b_2\,^2 + c_2\,2}}$ .......... (xiii)
- If two lines are perpendicular, $a_1a_2 + b_1b_2 + c_1c_2 = 0$
- Two lines will be parallel if $\frac{a_1}{a_2}$ = $\frac{b_1}{b_2}$ = $\frac{c_1}{c_2}$
