Overview of Three Dimensional Geometry

• Distance between P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):

\[PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\]​

• Distance of point (x, y, z) from origin

\[OP=\sqrt{x^2+y^2+z^2}\]

For points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) divided in ratio m₁ : m₂

(a) Internal Division

\[\left(\frac{m_1x_2+m_2x_1}{m_1+m_2},\frac{m_1y_2+m_2y_1}{m_1+m_2},\frac{m_1z_2+m_2z_1}{m_1+m_2}\right)\]

(b) External Division

\[\left(\frac{m_1x_2-m_2x_1}{m_1-m_2},\frac{m_1y_2-m_2y_1}{m_1-m_2},\frac{m_1z_2-m_2z_1}{m_1-m_2}\right)\]

(c) Mid-Point Formula

\[\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)\]

• XY-plane → z = 0

• YZ-plane → x = 0

• ZX-plane → y = 0

The angle between two skew lines is the angle between two intersecting lines drawn from any point parallel to each of the given lines. 

If a, b, c are three numbers proportional to the actual direction cosines l, m, n of a line, then the numbers a, b, c are called direction ratios (d.r.s.) of the line.

\[l=\frac{a}{\sqrt{a^2+b^2+c^2}},\quad m=\frac{b}{\sqrt{a^2+b^2+c^2}},\quad n=\frac{c}{\sqrt{a^2+b^2+c^2}}\]

If a line joins A(x₁,y₁,z₁) and B(x₂,y₂,z₂), then the direction ratios are: \[(x_2-x_1,y_2-y_1,z_2-z_1)\]

If (l₁,m₁,n₁) and (l₂,m₂,n₂) are the direction cosines of two lines, then the angle θ between them is given by

\[\cos\theta=l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}=\Sigma l_{1}l_{2}\]

Angle in Terms of Direction Ratios:

If the direction ratios of two lines are proportional to
(a₁,b₁,c₁) and (a₂,b₂,c₂) then:

\[\cos\theta=\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\]

\[\sin\theta=\frac{\sqrt{\left(a_1b_2-a_2b_1\right)^2+\left(b_1c_2-b_2c_1\right)^2+\left(c_1a_2-c_2a_1\right)^2}}{\sqrt{\Sigma a_1^2}\sqrt{\Sigma a_2^2}}\]

\[\cos\theta=\pm\frac{a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}}{\sqrt{\Sigma a_{1}^{2}}\sqrt{\Sigma a_{2}^{2}}}\]

\[\tan\theta=\frac{\sqrt{\Sigma\left(a_{1}b_{2}-a_{2}b_{1}\right)^{2}}}{\Sigma a_{1}a_{2}}\]

A straight line in space is uniquely determined if

1. It passes through a given point and has a given direction;
2. It passes through two given points. 

1. Symmetric (Standard) Form of a Line

If a line passes through (x₁,y₁,z₁) and has direction cosines (l,m,n) then its equation is:

\[\frac{x-x_{1}}{l}=\frac{y-y_{1}}{m}=\frac{z-z_{1}}{n}=r\]

2. Parametric Form (Coordinates of any Point)

\[x=x_1+lr,\quad y=y_1+mr, \quad z=z_1+nr\]

3. Line with Given Direction Ratios

\[\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\]

\[\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}\]

If a line passes through a point whose position vector is \[\vec{a}\] and is parallel to a given vector \[\vec{b}\], then its vector equation is:

\[\vec{r}=\vec{a}+\lambda\vec{b}\]

Where λ is a scalar parameter.

Line Through the Origin:

\[\vec{r}=\lambda\vec{b}\]

If a straight line passes through two points whose position vectors are \[\vec{a}\] and \[\vec{b}\], then the vector equation of the line is:

\[\vec{r}=\vec{a}+\lambda(\vec{b}-\vec{a})\]

or

\[\vec{r}=(1-\lambda)\vec{a}+\lambda\vec{b}\]

1. Cartesian → Vector Form

If the Cartesian equation of a line is:

\[\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\]

Then its vector form is:

\[\vec{r}=(x_1\hat{i}+y_1\hat{j}+z_1\hat{k})+\lambda(a\hat{i}+b\hat{j}+c\hat{k})\]

2. Vector → Cartesian Form

If the vector equation of a line is:

\[\vec{r}=\vec{a}+\lambda\vec{m}\]

Then the Cartesian form is:

\[\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\]

Vector Form:

\[\cos\theta=\frac{\vec{b}_1\cdot\vec{b}_2}{|\vec{b}_1||\vec{b}_2|}\]

Cartesian form:

\[\cos\theta=\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\]

Perpendicular condition:

\[a_1a_2+b_1b_2+c_1c_2=0\]

Parallel Condition:

\[\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\]

Coplanar:

Two straight lines are coplanar if they are either parallel or intersecting.

Skew Lines:

Two straight lines (in space) which are neither parallel nor intersecting are called skew lines.

Line of shortest distance: 

Let l₁,l₂ be two skew lines, then there is one and only one line (say l₃) which is perpendicular to both 1₁ and l₂. The line l₃ is known as the line of shortest distance. 

Shortest distance (S.D.):

 Let the line of shortest distance l₃ meet the given skew lines 1₁ and l₂ in points P and Q, respectively. Then \[\mid PQ\mid\] is the shortest distance between 1₁ and l₂.

S.D. between lines \[\begin{array}
{rcl}\vec{r} & = & a_1+\lambda\vec{b_1} & \mathrm{and} & \vec{r}=a_2+\mu\vec{b_2} & \mathrm{is}
\end{array}\]

\[\frac{\left(\overrightarrow{a}_{2}-\overrightarrow{a}_{1}\right)\cdot\left(\overrightarrow{b}_{1}\times\overrightarrow{b}_{2}\right)}{\left|\overrightarrow{b}_{1}\times\overrightarrow{b}_{2}\right|}=\frac{\left|\overrightarrow{a}_{2}-\overrightarrow{a}_{1},\overrightarrow{b}_{1},\overrightarrow{b}_{2}\right|}{\left|\overrightarrow{b}_{1}\times\overrightarrow{b}_{2}\right|}.\]

If the lines are: \[l_1:\vec{r}=\vec{a}_1+\lambda\vec{b}\] and \[l_2:\vec{r}=\vec{a}_2+\lambda\vec{b}\]

Then the shortest distance between them is: 

\[\mathrm{S.D.}=\frac{|\vec{b}\times(\vec{a}_2-\vec{a}_1)|}{|\vec{b}|}\]

A plane is a surface such that if two points are taken in it, the straight line joining them lies wholly in the surface. 

General Equation of a Plane:

\[ax+by+cz+d=0\]

Plane Passing Through the Origin:

\[ax+by+cz=0\]

If the plane passes through (x₁,y₁,z₁) then:

\[a(x-x_1)+b(y-y_1)+c(z-z_1)=0\]

If the plane passes through (x₁,y₁,z₁) (x₂​,y₂​,z₂​),(x₃​,y₃​,z₃​), then its equation is:

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

If

• p = length of the perpendicular from the origin to the plane

• (l,m,n) = direction cosines of the normal to the plane

Then the equation of the plane is:

\[lx+my+nz=p\]

\[\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\]

If the general equation of a plane is:

ax + by + cz + d = 0

Rewrite as:

ax + by + cz = −d

Then divide throughout by −d, to get:

\[\frac{x}{\frac{-d}{a}}+\frac{y}{\frac{-d}{b}}+\frac{z}{\frac{-d}{c}}=1\]

\[\vec{r}\cdot\hat{n}=p\]

If the plane passes through the origin:

\[\vec{r}\cdot\hat{n}=0\]

Corresponding Cartesian form:

\[lx+my+nz=p\]

Vector form:

\[(\vec{r}-\vec{a})\cdot\vec{n}=0\]

Cartesian form:

\[a(x-x_1)+b(y-y_1)+c(z-z_1)=0\]

Vector form:

If planes are: \[\vec{r}\cdot\vec{n}_1=q_1,\quad\vec{r}\cdot\vec{n}_2=q_2\] then:

\[\cos\theta=\frac{\vec{n}_1\cdot\vec{n}_2}{|\vec{n}_1||\vec{n}_2|}\]

Cartesian Form:

\[\cos\theta=\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\]

Vector Form:

\[\sin\theta=\frac{\overrightarrow{b}.\overrightarrow{n}}{|\overrightarrow{b}|.|\overrightarrow{n}|}\]

Cartesian Form:

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

Vector form:

If the planes are: \[\vec{r}\cdot\vec{n}_1=d_1\quad\mathrm{and}\quad\vec{r}\cdot\vec{n}_2=d_2\]

Then the plane through their intersection is:

\[(\vec{r}\cdot\vec{n}_1-d_1)+\lambda(\vec{r}\cdot\vec{n}_2-d_2)=0\]

Vector form:

\[\frac{|\vec{a}\cdot\vec{n}-d|}{|\vec{n}|}\]

Cartesian form:

\[\frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}\]

• Definition: Direction cosines are the cosines of the angles a line makes with the X, Y, Z axes.

• Notation: l = cos⁡α,  m = cos⁡β,  n = cos⁡γ

• \[l^2+m^2+n^2=1\]