Revision: Three - Dimensional Geometry Mathematics ISC (Science) ISC Class 12 CISCE

The two mutually perpendicular number lines intersecting each other at their zeroes are called rectangular axes or coordinate axes, or axes of reference. 

The position of a point in a plane is expressed by a pair of numbers, one concerning the x-axis and the other concerning the y-axis. called co-ordinates. 

• x → distance from y-axis (abscissa)

• y → distance from x-axis (ordinate)

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).

\[ \boxed{\,l^{2} + m^{2} + n^{2} = 1\,} \]

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:

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:

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

If the lines are \[ \vec{r} = \vec{a}_{1} + \lambda\vec{b}_{1} \] and \[ \vec{r} = \vec{a}_{2} + \mu\vec{b}_{2}, \]

then \[ \vec{a}_{1}, \vec{a}_{2} \] are position vectors of points on the lines and \[ \vec{b}_{1}, \vec{b}_{2} \] are their direction vectors.

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:

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

Vector Form:

\[\cos\theta=\left|\frac{\overline{\mathbf{n₁}}.\overline{\mathbf{n₂}}}{\left|\overline{\mathbf{n₁}}\right|.\left|\overline{\mathbf{n₂}}\right|}\right|\]

Cartesian Form:

\[\cos\theta=\left|\frac{\mathrm{a}_{1}\mathrm{a}_{2}+\mathrm{b}_{1}\mathrm{b}_{2}+\mathrm{c}_{1}\mathrm{c}_{2}}{\sqrt{\mathrm{a}_{1}^{2}+\mathrm{b}_{1}^{2}+\mathrm{c}_{1}^{2}}\sqrt{\mathrm{a}_{2}^{2}+\mathrm{b}_{2}^{2}+\mathrm{c}_{2}^{2}}}\right|\]

Vector Form:

\[\sin\theta=\left|\frac{\overline{\mathbf{b}}.\overline{\mathbf{n}}}{\left|\overline{\mathbf{b}}\right|.\left|\overline{\mathbf{n}}\right|}\right|\]

Cartesian Form:

\[\mathrm{sin}\theta=\frac{\mathrm{aa}_{1}+\mathrm{bb}_{1}+\mathrm{cc}_{1}}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}}\sqrt{\mathrm{a}_{1}^{2}+\mathrm{b}_{1}^{2}+\mathrm{c}_{1}^{2}}}\]

Vector Form:

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

Cartesian Form:

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

Sign Convention

• Right of y-axis → +x

• Left of y-axis → −x

• Above x-axis → +y

• Below x-axis → −y

Standard Line Results

• x = 0 → y-axis

• y = 0 → x-axis

• x = a → line parallel to the y-axis

• y = b → line parallel to the x-axis

Quadrant Reminder

• 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 \[(x_2 - x_1, y_2 - y_1, z_2 - z_1)\].

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

Equation of a Plane in Intercept form:

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

Distance of the Plane from Origin is

\[d=\frac{1}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}}}\]

Condition for Perpendicularity:

\[\overline{\mathbf{b}}=\lambda\overline{\mathbf{n}}\], λ is a parameter

\[\frac{\mathbf{a}_{1}}{\mathbf{a}}=\frac{\mathbf{b}_{1}}{\mathbf{b}}=\frac{\mathbf{c}_{1}}{\mathbf{c}}\]

Condition for Parallelism:

The line is parallel to the plane, if

\[\overline{\mathbf{b}}.\overline{\mathbf{n}}=0\]

aa₁ + bb₁ + cc₁ = 0