CBSE Class 12 Mathematics Three Dimensional Geometry Notes Set 01

BASIC CONCEPTS

1. Distance between two given points \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) is \[ |PQ| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}. \]

2. Direction ratios of a line joining the points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) are \( x_2 - x_1, \, y_2 - y_1, \, z_2 - z_1 \).

3. Angle between two lines, whose direction ratios are \( a_1, b_1, c_1 \) and \( a_2, b_2, c_2 \) is given by \[ \cos \theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \cdot \sqrt{a_2^2 + b_2^2 + c_2^2}}. \] (i) If lines are perpendicular, then \( a_1a_2 + b_1b_2 + c_1c_2 = 0 \).
(ii) If lines are parallel, then \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \).

4. Vector equation of a straight line passing through a fixed point with the position vector \( \vec{a} \) and parallel to a given vector \( \vec{b} \) is \( \vec{r} = \vec{a} + \lambda \vec{b} \), where \( \lambda \) is a parameter and \( \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \).

5. Cartesian equation (symmetrical form) of the straight line passing through a fixed point \( (x_1, y_1, z_1) \) having the direction ratios \( a, b, c \) is given by \( \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \)

6. The parametric equations of the line \( \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \) are \( x = x_1 + a\lambda, \, y = y_1 + b\lambda, \, z = z_1 + c\lambda \), where \( \lambda \) is a parameter.

7. The coordinates of any point on the line \( \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \) are \( (x_1 + a\lambda, y_1 + b\lambda, z_1 + c\lambda) \), where \( \lambda \in \mathbb{R} \).

8. Equation of straight line passing through the point \( (x_1, y_1, z_1) \) having direction cosines \( l, m, n \) is \( \frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n} \)

9. Vector equation of two straight lines passing through two given points with position vector \( \vec{a} \) and \( \vec{b} \) is \( \vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a}) \), where \( \lambda \) is a parameter.

10. Cartesian equation of a straight line passing through two given points \( A (x_1, y_1, z_1) \) and \( B (x_2, y_2, z_2) \) is given by \( \frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1} \)

11. Angle between two straight lines: Angle between two straight lines whose vector equations are \( \vec{r} = \vec{a}_1 + \lambda \vec{b}_1 \) and \( \vec{r} = \vec{a}_2 + \lambda \vec{b}_2 \), is equal to the angle between \( \vec{b}_1 \) and \( \vec{b}_2 \), because \( \vec{b}_1 \) and \( \vec{b}_2 \) are parallel vector to the lines \( \vec{r} = \vec{a}_1 + \lambda \vec{b}_1 \) and \( \vec{r} = \vec{a}_2 + \lambda \vec{b}_2 \) respectively.
If \( \theta \) is angle between both lines, then \[ \cos \theta = \left| \frac{\vec{b}_1 \cdot \vec{b}_2}{|\vec{b}_1| |\vec{b}_2|} \right| \] (i) If \( \vec{b}_1 \cdot \vec{b}_2 = 0 \), then \( \cos \theta = 0 \)
\( \implies \) \( \theta = 90^\circ \)
\( \implies \) \( \vec{b}_1 \perp \vec{b}_2 \)
\( \implies \) both lines are perpendicular to each other.
(ii) If \( \vec{b}_1 = \lambda \vec{b}_2 \), then
\( \implies \) \( \cos \theta = \frac{\lambda \vec{b}_2 \cdot \vec{b}_2}{|\lambda \vec{b}_2| |\vec{b}_2|} = \frac{\lambda |\vec{b}_2|^2}{\lambda |\vec{b}_2|^2} \)
\( \implies \) \( \cos \theta = 1 \)
\( \implies \) \( \theta = 0^\circ \)
\( \implies \) \( \vec{b}_1 \parallel \vec{b}_2 \)
\( \implies \) Both lines are parallel to each other.

12. Shortest distance between two lines: Let \( l_1 \) and \( l_2 \) be two skew lines given by \( \vec{r} = \vec{a}_1 + \lambda \vec{b}_1 \) and \( \vec{r} = \vec{a}_2 + \lambda \vec{b}_2 \) respectively, where \( a_1 \) and \( a_2 \) are position vectors of points on \( l_1 \) and \( l_2 \) then shortest distance between two given points is given by \[ \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|} \] Note: If two lines are intersecting, then shortest distance between them is zero, i.e., \( (\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = 0 \)

13. Shortest distance between two parallel lines: Let \( l_1 \) and \( l_2 \) be two parallel lines given by \( \vec{r} = \vec{a}_1 + \lambda \vec{b} \) and \( \vec{r} = \vec{a}_2 + \lambda \vec{b} \) respectively. Then shortest distance between them is \( \frac{|\vec{b} \times (\vec{a}_2 - \vec{a}_1)|}{|\vec{b}|} \)

14. Shortest distance between two skew lines in cartesian form: Let \( \frac{x - x_1}{a_1} = \frac{y - y_1}{b_1} = \frac{z - z_1}{c_1} \) and \( \frac{x - x_2}{a_2} = \frac{y - y_2}{b_2} = \frac{z - z_2}{c_2} \) are two skew lines, then shortest distance between them is given by \[ \begin{vmatrix} x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} \] Note: If \( \begin{vmatrix} x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} = 0 \). Then lines are intersecting.

15. Equation of a plane passing through given point, whose position vector is \( \vec{a} \) and perpendicular to a given vector \( \vec{n} \), is \( \vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n} \) or \( (\vec{r} - \vec{a}) \cdot \hat{n} = 0 \).

16. Cartesian equation of plane passing through a given point \( (x_1, y_1, z_1) \) and perpendicular to the normal, whose direction ratios are \( a, b, c \) respectively is given by \( a(x - x_1) + b(y - y_1) + c(z - z_1) = 0 \)

17. Equation of a plane in normal form:
(i) When a unit vector \( (\hat{n}) \) perpendicular (normal) to the plane is given and its perpendicular distance \( d \) from the origin is also given, then the equation of plane is \( \vec{r} \cdot \hat{n} = d \).
(ii) If unit vector \( \hat{n} = l\hat{i} + m\hat{j} + n\hat{k} \) where \( l, m, n \) are direction cosines and \( p \), the perpendicular distance from origin to normal are given, then equation of the plane is \( lx + my + nz = p \).

18. Angle between line and plane: Angle between line \( \vec{r} = \vec{a} + \lambda \vec{b} \) and plane \( \vec{r} \cdot \vec{n} = d \) is complementary to the angle between line and the normal to the plane.
Let \( \theta \) be the angle between line and plane and \( \phi \), the angle between line and normal of plane.
Now, \( \cos \phi = \left| \frac{\vec{b} \cdot \vec{n}}{|\vec{b}| |\vec{n}|} \right| \)
\( \because \theta = 90 - \phi \) or \( \phi = 90 - \theta \)
\( \therefore \cos (90 - \theta) = \left| \frac{\vec{b} \cdot \vec{n}}{|\vec{b}| |\vec{n}|} \right| \)

\( \implies \) \( \sin \theta = \left| \frac{\vec{b} \cdot \vec{n}}{|\vec{b}| |\vec{n}|} \right| \)
(i) If \( \vec{b} = \lambda \vec{n} \), then \( \sin \theta = \left| \frac{\lambda \vec{n} \cdot \vec{n}}{|\lambda \vec{n}| |\vec{n}|} \right| = 1 \)

\( \implies \) \( \theta = 90^\circ \)

\( \implies \) line is perpendicular to the plane.
(ii) If \( \vec{b} \cdot \vec{n} = 0 \)

\( \implies \) \( \sin \theta = 0 \)

\( \implies \) \( \theta = 0^\circ \)

\( \implies \) line is parallel to the plane.

19. Angle between two planes: The angle between two planes is defined as the angle between their normals.
Case I. If \( \theta \) be the angle between normals \( \vec{n}_1 \) and \( \vec{n}_2 \) of the planes \( \vec{r} \cdot \vec{n}_1 = d_1 \) and \( \vec{r} \cdot \vec{n}_2 = d_2 \) respectively, then
\[ \cos \theta = \left| \frac{\vec{n}_1 \cdot \vec{n}_2}{|\vec{n}_1| |\vec{n}_2|} \right| \]
Note: (i) If \( \vec{n}_1 \cdot \vec{n}_2 = 0 \) then given planes are perpendicular.
(ii) If \( \vec{n}_1 = \lambda \vec{n}_2 \), then both planes are parallel.
(iii) The angle between two planes is always taken as acute angle.
Case II. If \( a_1x + b_1y + c_1z + d_1 = 0 \) and \( a_2x + b_2y + c_2z + d_2 = 0 \) be two planes and \( \theta \) is the angle between them, where \( a_1, b_1, c_1 \) and \( a_2, b_2, c_2 \) are direction ratios of normals to the planes, then
\[ \cos \theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \cdot \sqrt{a_2^2 + b_2^2 + c_2^2}} \]
(i) If \( a_1a_2 + b_1b_2 + c_1c_2 = 0 \), then planes are perpendicular to each other.
(ii) If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \) then given planes are parallel to each other.

Determination of plane under given conditions:

20. An equation of first degree in \( x, y, z \) of the form \( ax + by + cz + d = 0 \) where at least one of \( a, b, c \) is non-zero real number, i.e., \( a^2 + b^2 + c^2 \neq 0 \) represents a plane.

21. Equation of plane in intercept form: \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \) where \( a, b, c \) are intercepts made by the plane on the x-axis, y-axis and z-axis respectively.

22. Equation of plane passing through three given points:
Case I. Vector equation of the plane passing through three given points having position vector \( \vec{a} \), \( \vec{b} \) and \( \vec{c} \) is given by
\[ [(\vec{r} - \vec{a}) \, (\vec{b} - \vec{a}) \, (\vec{c} - \vec{a})] = 0 \]
or
\[ (\vec{r} - \vec{a}) \cdot ((\vec{b} - \vec{a}) \times (\vec{c} - \vec{a})) = 0 \]
Case II. Cartesian equation of the plane passing through points \( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \) and \( C(x_3, y_3, z_3) \) is given by
\[ \begin{vmatrix} x - x_1 & y - y_1 & z - z_1 \\ x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \end{vmatrix} = 0 \]

23. Condition for coplanarity of two lines:
Case I. When lines are in vector form:
(i) Let \( \vec{r} = \vec{a}_1 + \lambda \vec{b}_1 \) and \( \vec{r} = \vec{a}_2 + \lambda \vec{b}_2 \) be two lines then these lines are coplanar if
\[ [\vec{a}_2 - \vec{a}_1 \, \vec{b}_1 \, \vec{b}_2] = 0 \]
(ii) Equation of plane containing these two lines is
\( (\vec{r} - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = 0 \) or \( (\vec{r} - \vec{a}_2) \cdot (\vec{b}_1 \times \vec{b}_2) = 0 \).
Case II. When lines are in cartesian form:
(i) Let \( \frac{x - x_1}{a_1} = \frac{y - y_1}{b_1} = \frac{z - z_1}{c_1} \) and \( \frac{x - x_2}{a_2} = \frac{y - y_2}{b_2} = \frac{z - z_2}{c_2} \) be the two lines then these lines are coplanar iff
\[ \begin{vmatrix} x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} = 0 \]
(ii) Equation of plane containing these lines is
\( \frac{x - x_1}{b_1c_2 - b_2c_1} = \frac{y - y_1}{a_2c_1 - a_1c_2} = \frac{z - z_1}{a_1b_2 - a_2b_1} \) or \( \frac{x - x_2}{b_1c_2 - b_2c_1} = \frac{y - y_2}{a_2c_1 - a_1c_2} = \frac{z - z_2}{a_1b_2 - a_2b_1} \)
(iii) The length of perpendicular from a point having position vector \( \vec{a} \) to the plane \( \vec{r} \cdot \vec{n} = d \) is given by \( \frac{| \vec{a} \cdot \vec{n} - d |}{|\vec{n}|} \)
(iv) The length of perpendicular from a point \( P(x_1, y_1, z_1) \) to the plane \( ax + by + cz + d = 0 \) is given by
\[ \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}} \]