Students can Download Chapter 11 Three Dimensional Geometry Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

## Kerala Plus Two Maths Notes Chapter 11 Three Dimensional Geometry

Introduction

To refer a point in space we require a third axis (say z-axis) which leads to the concept of three dimensional geometry. In this chapter we study the concept of direction cosines, direction ratios, equation of a line and a plane, angle between two lines and two planes, angle between a line and a plane, shortest distance between two skew lines, distance of a point from a plane.

Basic concepts

I. Direction cosines and direction ratios

Consider a directed line passing through the origin makes angles α, β, and γ with the positive

direction x-axis, y-axis, and z-axis. Then α, β, and γ are called direction angles. The cosine of α, β, and γ are called direction cosines. Generally cos α = l, cos β = m and cos γ = n . Any scalar multiple of direction cosines are called direction ratios.

1. If (a, b, c) is the coordinate of a point P then a,b,c is a direction ratio of the directed line passing along P and origin. Direction cosines will be

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

3. Direction ratios of a line segment passing through two points(x_{1}, y_{1}, z_{1}) and(x_{2}, y_{2}, z_{2}) is

x_{2} – x_{1}, y_{2} – y_{1}, z_{1} – z_{2}

4. The angle between two lines having direction ratios a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2} is

5. If direction ratios are proportional then the lines a, b, c, are parallel.ie; \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\).

6. If a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0 then the two lines are perpendicular.

II. Line in Space

Equation of line when a point and parallel direction ratios are given:

1. Vector equation:

\(\bar{r}=\bar{a}+\lambda \bar{b}\), where \(\bar{a}\) is a point, \(\bar{b}\) is a parallel vector and λ is a parameter, for different values of λ we get parallel lines.

2. Cartesian equation:

where (x_{1}, y_{1}, z_{1}) is a point and a, b, c is a parallel direction ratios.

Equation of line when two points are given:

1. Vector equation:

\(\bar{r}=\bar{a}+\lambda(\bar{b}-\bar{a})\), where \(\bar{a}\) and \(\bar{b}\) are points and λ is a parameter, for different values of λ we get parallel lines.

2. Cartesianequation:

where(x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) are two points.

Angle between two lines:

1. Vector form:

be two lines and θ be the angle between them, then cosθ =

- If parallel \(\overline{b_{1}}=k \overline{b_{2}}\), k scalar.
- If perpendicular \(\overline{b_{1}} \overline{b_{2}}\) = 0.

2. Cartesian form:

be two lines and θ be the angle between them, then

- If parallel \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)
- If perpendicular a
_{1}a_{2}+ b_{1}b_{2}+ c_{1}c_{2}= 0.

Shortest distance between skew lines:

Lines which are neither interesting nor parallel are known as skew lines. Shortest distance between two skew lines is

1. Vector form:

be two skew lines, then d =

2. Cartesian form:

3. Distance between parallel lines,

III. Plane in space

Normal Form:

1. Vector Equation:

\(\bar{r}\).\(\hat{n}\) = d, where \(\hat{n}\) is a unit vector perpendicular to the Plane, and d is the perpendicular distance of the Plane from the origin. The general vector equation of a Plane is \(\bar{r} \bar{m}=d\), where \(\bar{m}\) is any vector perpendicular to the plane and cfis a constant.

2. Cartesian equation:

lx + my + nz = d, where l, m, n are direction cosines perpendicular to the Plane and dis the perpendicular distance of the Plane from the origin. The general cartesian equation of a Plane is ax + by + cz = d, where a, b, c are direction ratios perpendicular to the plane, and d is a constant.

Equation of plane when a point and a perpendicular vector is given:

1. Vector Equation:

\((\bar{r}-\bar{a}) \bar{m}=d\), where \(\bar{m}\) is a vector perpendicular to the Plane and \(\bar{a}\) is a point on the plane.

2. Cartesianequation:

a(x – x_{1}) + b(y – y_{1}) + c(z – z_{1}) = d, where a, b, c are direction ratios perpendicular to the plane and (x_{1}, y_{1}, z_{1}) is a point on the plane.

Equation of a plane passing through three non-collinear points:

1. Vector Equation:

\((\bar{r}-\bar{a}) \cdot[(\bar{b}-\bar{a}) \times(\bar{c}-\bar{a})]=0\), where \(\bar{a}, \bar{b}, \bar{c}\) are points on the plane.

2. Cartesian equation:

Where, (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, z_{3}) are points on the plane.

3. Intercept form of the equation of a Plane:

Let a, b, c are the x-intercept, y-intercept and z- intercept made by a plane, then the equation of x y z such a Plane is \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\).

4. Angle between two Planes:

Angle between two planes is same as the angle between there perpendicular vectors.

5. Vector Form:

be two Planes and θ be the angle between them, then cos θ

6.

(i) if parallel \(\overline{m_{1}}=k \overline{m_{2}}\), k scalar.

(ii) if perpendicular \(\bar{m}_{1} \overline{m_{2}}\) = 0

7. Cartesian form:

a_{1}x + b_{1}y + c_{1}z = d_{1}, a_{2}x + b_{2}y + c_{2}z = d_{2} be two lines and θ be the angle between them, then

- If parallel \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)
- If perpendicular a
_{1}a_{2}+ b_{1}b_{2}+ c_{1}c_{2}= 0

Angle between a line and a Plane:

Plane passing through the intersection of two given planes:

The equation of family of Planes passing through the intersection of the Planes a_{1}x + b_{1}y + c_{1}z = d_{1} and a_{2}x + b_{2}y + c_{2}z = d_{2} is a_{1}x + b_{1}y + c_{1}z – d_{1} + λ(a_{2}x + b_{2}y + c_{2}z – d) = 0.

Distance of a point from a Plane:

The perpendicular distance of the point (x_{1}, y_{1}, z_{1}) from a Plane ax + by + cz = d is

The distance between parallel Planes ax + by + cz = d_{1} and ax + by + cz = d_{2} is