Revision: Vectors and Three-dimensional Geometry >> Three - Dimensional Geometry Maths Commerce (English Medium) Class 12 CBSE

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

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. 

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.

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:

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

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.

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

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

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:

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

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:

\[ax+by+cz=0\]

• 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

• \[l^2+m^2+n^2=1\]

When slope and y-intercept are given

y = mx + c​

• m = slope

• c = y-intercept (value of y when x = 0)

When two points are given

\[\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}\]​​

When the slope and one point are given

y − y₁ = m(x − x₁)

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

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

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

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

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:

\[lx+my+nz=p\]

Vector form:

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

Cartesian form:

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

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

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: 

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

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

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

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:

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

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:

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

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

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

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:

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

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

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

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

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:

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

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:

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

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

• 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

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:

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