# Plus Two Maths Notes Chapter 11 Three Dimensional Geometry

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## 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. l2 + m2 + n2 = 1

3. Direction ratios of a line segment passing through two points(x1, y1, z1) and(x2, y2, z2) is
x2 – x1, y2 – y1, z1 – z2

4. The angle between two lines having direction ratios a1, b1, c1 and a2, b2, c2 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 a1a2 + b1b2 + c1c2 = 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 (x1, y1, z1) 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(x1, y1, z1) and (x2, y2, z2) 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 a1a2 + b1b2 + c1c2 = 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 – x1) + b(y – y1) + c(z – z1) = d, where a, b, c are direction ratios perpendicular to the plane and (x1, y1, z1) 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, (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) 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:
a1x + b1y + c1z = d1, a2x + b2y + c2z = d2 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 a1a2 + b1b2 + c1c2 = 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 a1x + b1y + c1z = d1 and a2x + b2y + c2z = d2 is a1x + b1y + c1z – d1 + λ(a2x + b2y + c2z – d) = 0.

Distance of a point from a Plane:
The perpendicular distance of the point (x1, y1, z1) from a Plane ax + by + cz = d is

The distance between parallel Planes ax + by + cz = d1 and ax + by + cz = d2 is