Odisha State Board CHSE Odisha Class 12 Math Notes Chapter 13 Three Dimensional Geometry will enable students to study smartly.

## CHSE Odisha 12th Class Math Notes Chapter 13 Three Dimensional Geometry

**Important Formulae:**

Distance Formula:

The distance between two points (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2})

= \(\sqrt{\left(x_1-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}\)

Division Formula:

(i) Internal division:

If P (x, y, z) divides the line segment joining, A (x_{1}, y_{1}, z_{1}) and B (x_{2}, y_{2}, z_{2}) into the ratio m : n internally then

Remark:

(i) If P (x, y, z) divides the line segment joining the points A (x_{1}, y_{1}, z_{1}) and B (x_{2}, y_{2}, z_{2}) into the ratio λ : 1 then

(ii) Co-ordinates of the mid-point of the line segment joining the points (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) are

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

Direction Cosines:

Suppose that a straight line makes angles α, β, γ with the positive directions of x-axis, y-axis and z-axis respectively.

Then direction cosines of the line are < cos α, cos β, cos γ >

We denote l = cos α, m = cos β and n = cos γ

Then l^{2} + m^{2} + n^{2} = 1

Direction Ratios:

The direction ratios of a straight line are proportional to direction cosines.

If < a, b, c > are d. rs. and < l, m, n > are d.cs then

Direction ratios of a line segment 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} >

The projection of a line segment joining the points A (x_{1}, y_{1}, z_{1}) and B (x_{2}, y_{2}, z_{2}) onto the line ‘L’ with d.cs. < l, m, n >

= l (x_{2} – x_{1}) + m (y_{2} – y_{1}) + n (z_{2} – z_{1})

Angle between two lines:

Angle between two lines with d.cs.

< l_{1}, m_{1}, n_{1} > and < l_{2}, m_{2}, n_{2} > is given by cos θ = l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2}

(i) Two lines are parallel if their d.cs. are equal or d.r.s. are proportional.

(ii) Two lines are perpendicular if l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} = 0

**Plane **

Important formulae:

1. The general equation of the plane is ax + by + cz + d = 0

2. Equation of the plane passing through a poing (x_{1}, y_{1}, z_{1}) and having l, m, n direction cosines of the normal to the plane is l (x – x_{1}) + m (y – y_{1}) + n (z – z_{1}) = 0

3. Equation of the plane in intercept form is \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\) = 1

where a, b, c are the intercepts from the axes.

4. Equation of the plane in normal form is lx + my + nz = p

where < l, m, n > are d.cs of the normal and p is the length of the normal.

5. Equation of the plane passing through three points.

(x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, z_{3}) is

\(\left|\begin{array}{rrr}

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{array}\right|\) = 0

6. (i) Angle between two planes is the angle between their normals.

(ii) If two planes are

a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and

a_{2}x + b_{2}y + c_{2}z + d_{2} = 0

then the direction ratios of their normal are < a_{1}, b_{1}, c_{1} > and < a_{2}, b_{2}, c_{2} >

(iii) If θ is the angle between two planes then

cos θ = \(\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \cdot \sqrt{a_2^2+b_2^2+c_2^2}}\)

(iv) Two planes a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 are parallel if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\).

(v) The above two planes are perpendicular if a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0.

7. The distance of a point (x_{1}, y_{1}, z_{1}) from a plane ax + by + cz + d = 0 is

\(\left|\frac{a x_1+b y_1+c z_1+d}{\sqrt{a^2+b^2+c^2}}\right|\)

8. Equations of the planes bisecting the angle between two planes

a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and

a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 are

\(\frac{a_1 x+b_1 y+c_1 z+d_1}{\sqrt{a_1^2+b_1^2+c_1^2}}=\pm \frac{a_2 x+b_2 y+c_2 z+d_2}{\sqrt{a_2^2+b_2^2+c_2^2}}\)

**The Straight Line**

Important formulae:

1. Unsymmetrical From:

The joint equation of two planes represent a stright line. Thus the equation of a straight line in unsymmetrical form a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0

2. Symmetrical Form:

Equation of a straight line through a point (x_{0}, y_{0}, z_{0}) and having d.c. < l, m, n > is

\(\frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}\)

3. Two-point Form:

The equation of a straight line passing through two points (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) is

\(\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}\)

4. Condition that a line will lie on a Plane:

The straight line

\(\frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}\) lie in a plane ax + by + cz + d = 0

if (i) al + bm + cn = 0

and (ii) ax_{0} + by_{0} + cz_{0} + d = 0

5. Condition for Two Lines to be Coplanar:

6. Angle between a line and a plane:

The angle between the line

7. Distance of a Point from a Line:

The distance of a point (x_{1}, y_{1}, z_{1}) from a line