Revision: Mathematics >> Three-dimensional Geometry CUET (UG) Three-dimensional Geometry

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:

An equation of the form ax + by + c = 0 represents a straight line and is known as a linear equation.

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:

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:

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

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:

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

From general form:

• Slope (m) = −a / b
• Y-intercept = −c / b

Vector Form:

\[\mathbf{d}=\frac{\left|\left(\overline{\mathbf{a}}.\overline{\mathbf{n}}\right)-\mathbf{p}\right|}{\left|\overline{\mathbf{n}}\right|}\]

Cartesian Form:

\[\mathbf{d}=\left|\frac{\mathbf{a}x_{1}+\mathbf{b}y_{1}+\mathbf{c}z_{1}+\mathbf{d}}{\sqrt{\mathbf{a}^{2}+\mathbf{b}^{2}+\mathbf{c}^{2}}}\right|\]

Vector:

Angle between two lines: \[\cos\theta=\left|\frac{\mathbf{b}_{1}\cdot\mathbf{b}_{2}}{|\mathbf{b}_{1}||\mathbf{b}_{2}|}\right|\]

If two lines are perpendicular: b₁ · b₂ = 0

If two lines are parallel: b₁ = λb₂

Cartesian: 

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

If two lines are perpendicular: a₁a₂ + b₁b₂ + c₁c₂ = 0

If two lines are parallel: \[\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\]

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

Position of a Point:

For line: ax₁ + by₁ + c

• If ax₁ + by₁ + c = 0 → Point lies on the line
• If ax₁ + by₁ + c < 0 → Point lies on one side (origin side)
• If ax₁ + by₁ + c > 0 → Point lies on other side

Vector Form:

Condition for coplanarity of two lines:

Two lines r = a₁ + λb₁ and r = a₂ + μb₂ are coplanar if

(a₁ − a₂) · (b₁ × b₂) = 0

Equation of the plane containing both lines:

\[\left(\overline{\mathbf{r}}-\overline{\mathbf{a}_1}\right).\left(\overline{\mathbf{b}_1}\times\overline{\mathbf{b}_2}\right)=\mathbf{0}\] or \[\left(\overline{\mathbf{r}}-\overline{\mathbf{a}_2}\right).\left(\overline{\mathbf{b}_1}\times\overline{\mathbf{b}_2}\right)=\mathbf{0}\]

Cartesian Form:

\[\begin{vmatrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ \mathbf{a}_1 & \mathbf{b}_1 & \mathbf{c}_1 \\ \mathbf{a}_2 & \mathbf{b}_2 & \mathbf{c}_2 \end{vmatrix}=0\]