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

If a directed line makes angles \[\alpha\], \[\beta\], and \[\gamma\] with the positive x-, y-, and z-axes respectively, then these are called the direction angles of the line.

The cosines of these angles are called the direction cosines of the line.

So, the direction cosines are written as (l, m, n).

Any three numbers proportional to the direction cosines of a line are called the direction ratios of the line.

If (a, b, c) are direction ratios, then:

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 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:

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

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.

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

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

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. 

If the direction cosines of the two lines are

\[(l_1, m_1, n_1)\] and \[(l_2, m_2, n_2)\], then:

If the direction ratios are \[(a_1, b_1, c_1)\] and \[(a_2, b_2, c_2)\], then:

If the direction ratios of two lines are:

First line: \[(a_1, b_1, c_1)\]

Second line: \[(a_2, b_2, c_2)\]

then the cosine of the angle \[\theta\] between them is:

Vector Form:

\[\mathbf{d}=\left|\frac{(\overline{\mathbf{b}}_{1}\times\overline{\mathbf{b}}_{2}).(\overline{\mathbf{a}}_{2}-\overline{\mathbf{a}}_{1})}{\left|\overline{\mathbf{b}}_{1}\times\overline{\mathbf{b}}_{2}\right|}\right|\]

Cartesian Form:

\[\mathbf{d}=\left|\frac{ \begin{vmatrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ \mathbf{a}_1 & \mathbf{b}_1 & \mathbf{c}_1 \\ \mathbf{a}_2 & \mathbf{b}_2 & \mathbf{c}_2 \end{vmatrix}}{\sqrt{\left(\mathbf{a}_1\mathbf{b}_2-\mathbf{a}_2\mathbf{b}_1\right)^2+\left(\mathbf{a}_1\mathbf{c}_2-\mathbf{a}_2\mathbf{c}_1\right)^2+\left(\mathbf{b}_1\mathbf{c}_2-\mathbf{b}_2\mathbf{c}_1\right)^2}}\right|\]

\[SD=\left|\frac{\left(a_{2}-a_{1}\right)\times b}{\left|b\right|}\right|\]

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:

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

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 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:

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

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

If the plane passes through the origin:

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

Corresponding Cartesian form:

\[lx+my+nz=p\]

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

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

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

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

Vector form:

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

Cartesian form:

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

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:

\[lx+my+nz=p\]

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

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

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

• 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

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

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

Vector form:

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

Cartesian form:

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

• Direction Cosines (DCs) of a line: \[(l, m, n) = (\cos \alpha, \cos \beta, \cos \gamma)\].

• Main identity: \[l^2 + m^2 + n^2 = 1\].

• Direction Ratios (DRs): Are proportional to DCs.

• Relation between DRs and DCs: If DRs are (a, b, c), then DCs are proportional to (a, b, c) divided by \[\sqrt{a^2 + b^2 + c^2}\].

• DRs for two points: For points \[P(x_1, y_1, z_1)\] and \[Q(x_2, y_2, z_2)\], DRs are \[(x_2 - x_1, y_2 - y_1, z_2 - z_1)\].

• Through point \(\vec a\) and parallel to \(\vec b\): \(\vec r = \vec a + \lambda \vec b\).

• Parametric form: \(x = x_1 + \lambda a,; y = y_1 + \lambda b,; z = z_1 + \lambda c\).

• Cartesian form: \(\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}\).

• Through two points: \(\vec r = \vec a + \lambda(\vec b-\vec a)\).

• The angle between two lines depends only on their directions.

• If lines do not pass through the origin, imagine parallel lines through the origin.

• The dot-product formula is the main method for solving these questions.

• In symmetric form, denominators give direction ratios.

• Zero dot product means perpendicular lines.

• Proportional direction ratios mean parallel lines.

• The required angle is generally the acute angle.

• Intersecting lines: SD = 0

• Parallel lines: \[SD = \frac{|(\vec{a}_2 - \vec{a}_1) \times \vec{b}|}{|\vec{b}|}\]

• Skew lines: \[SD = \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|}\]

• 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:

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