# Three Dimensional Geometry: Class 12 Mathematics NCERT Chapter 11

## Key Features of NCERT Material for Class 12 Maths Chapter 11 – Three Dimensional Geometry

In the last chapter 10, you learned about Vector Algebra. In this chapter, you will learn everything about Three Dimensional Geometry.

### Quick revision notes

Direction Cosines of a Line: If the directed line OP makes angles α, β, and γ with positive X-axis, Y-axis and Z-axis respectively, then cos α, cos β, and cos γ, are called direction cosines of a line. They are denoted by l, m, and n. Hence, l = cos α, m = cos β and n = cos γ. Also, sum of squares of direction cosines of a line is always 1,

i.e. l2 + m2 + n2 = 1 or cos2 α + cos2 β + cos2 γ = 1

Note: Direction cosines of a directed line are unique.

Direction Ratios of a Line: Number proportional to the direction cosines of a line, are called direction ratios of a line.

(i) If a, b and c are the direction ratios of a line, then  =  =

(ii) If a, b and care direction ratios of a line, then its direction cosines are

(iii) Direction ratios of a line PQ passing through the points P(x1, y1, z1) and Q(x2, y2, z2) are x2 – x1, y2 – y1 and z2 – z1 and direction cosines are

Note:

(i) Direction ratios of two lines which are parallel are proportional.

(ii) Direction ratios of a given line are not unique.

Straight line: A straight line is a curve, such that all the points on the line segment joining any two points of it lies on it.

Equation of a given Line through a Given Point and parallel to a given vector

Vector form where, = Position vector of a point through which the line is passing

= A vector parallel to a given line

Cartesian form

where, (x1, y1, z1) is the point through which the line is passing through and a, b, c are the direction ratios of the line.

If l, m, and n are the direction cosines of the line, then the equation of the line is

Remember point: Before we use the DR’s of a line, first we have to ensure that coefficients of x, y and z are unity with a positive sign.

#### Equation of Line Passing through Two Given Points

Vector forms:  , λ ∈ R, where a and b are the position vectors of the points through which the line is passing.

Cartesian form

where, (x1, y1, z1) and (x2, y2, z2) are the points through which the line is passing.

#### Angle between Two Lines

Vector form: Angle between the lines and is given as

Condition of Perpendicularity: Two lines are known to be perpendicular, when in vector form ; in cartesian form a1a2 + b1b2 + c1c2 = 0

or l1l2 + m1m2 + n1n2 = 0 [direction cosine form]

Condition that Two Lines are Parallel: Two lines are parallel, when in vector form ; in cartesian form

or

[direction cosine form]

Shortest Distance between Two Lines: Two non-parallel and non-intersecting straight lines, are called skew lines.

For skew lines, the line having the shortest distance will be perpendicular to both the lines.

Vector form: If the lines are and . Then, shortest distance

where , are position vectors of point through which the line is passing and , are the vectors in the direction of a line.

Cartesian form: If the lines are

Then, shortest distance,

Distance between two Parallel Lines: If two lines l1 and l2 are parallel, then they are coplanar. Let the lines be and , then the distance between parallel lines is

Note: If two lines are parallel, then they both have same DR’s.

Distance between Two Points: The distance between two points P (x1, y1, z1) and Q (x2, y2, z2) is given by

Mid-point of a Line: The mid-point of a line joining points A (x1, y1, z1) and B (x2, y2, z2) is given by

Plane: A plane is a surface such that a line segment joining any two points of it lies entirely on it. A straight line which is perpendicular to every line lying on a plane is called a normal to the plane.

#### Equations of a Plane in Normal form

Vector form: The equation of plane in normal form is given by , where is a vector which is normal to the plane.

Cartesian form: The condition of the plane is given by hatchet + by + cz = d, where a, b and c are the direction ratios of plane and d is the distance of the plane from inception.

Another condition of the plane is lx + my + nz = p, where l, m, and n are direction cosines of the perpendicular from inception and p is a distance of a plane from cause.

Note: If d is the distance from the inception and l, m and n are the direction cosines of the typical to the plane through the root, at that point the foot of the perpendicular is (ld, md, nd).

Condition of a Plane Perpendicular to a given Vector and Passing Through a given Point

Vector form: Let a plane goes through a guide A with position vector and perpendicular to the vector , at that point

This is the vector equation of the plane.

Cartesian form: Equation of plane going through point (x1, y1, z1) is given by

a (x – x1) + b (y – y1) + c (z – z1) = 0 where, a, b and c are the direction ratios of typical to the plane.

#### Condition of Plane Passing through Three Non-collinear Points

Vector form: If  , and are the position vectors of three given points, at that point equation of a plane passing through three non-collinear points is .

Cartesian form: If (x1, y1, z1) (x2, y2, z2) and (x3, y3, z3) are three non-collinear points, at that point the equation of the plane is

If above points are collinear, at that point

Equation of Plane in Intercept Form: If a, b and c are x-intercept, y-intercept and z-intercept, separately made by the plane on the coordinate axes, then equation of plane is

Equation of Plane Passing through the Line of Intersection of two given Planes x

Vector form: If equation of the planes are and , then equation of any plane passing through this given intersection of planes is

where, λ is a constant and determined from a given condition.

Cartesian form: If the condition of planes are a1x + b1y + c1z = d1 and a2x + b2y + c2z = d2, at that point condition of any plane going through the convergence of planes is a1x + b1y + c1z – d1 + λ (a2x + b2y + c2z – d2) = 0

where, λ is a steady and determined from a given condition.

#### Coplanarity of Two Lines

Vector form: If two lines and are coplanar, then

Angle between Two Planes: Let θ be the angle between two planes.

Vector form: If and are normals to the planes and θ be the angle between the planes and , at that point θ is the angle between the normals to the planes drawn from some regular points.

Note: The planes are perpendicular to one another, if and parallel, if

Cartesian form: If the two planes are a1x + b1y + c1z = d1 and a2x + b2y + c2z = d2, at that point

Note: Planes are perpendicular to one another, if a1a2 + b1b2 + c1c2 = 0 and planes are parallel, if

#### Distance of a Point from a Plane

Vector form: The distance of a point whose position vector is from the plane

Note:

(i) If the equation of the plane is of the form , where is normal to the plane, at that point the perpendicular distance is

(ii) The length of the perpendicular from starting point O to the plane [∵ = 0]

Cartesian form: The distance of the point (x1, y1, z1) from the plane Ax + By + Cz = D is

#### Angle between a Line and a Plane

Vector form: If the equation of line is and the equation of plane is , then the angle θ between the given line and the normal to the plane is

and thus the angle Φ between the line and the plane is given by 90° – θ,

i.e. sin(90° – θ) = cos θ

Cartesian form: If a, b and c are the DR’s of line and lx + my + nz + d = 0 be the equation of plane, at that point

In the event that a line is parallel to the plane, then al + bm + cn = 0 and on the off chance line is perpendicular to the plane, at that point

#### Remember Points

(i) If a line is parallel to the plane, at that point normal to the plane is perpendicular to the line. i.e. a1a2 + b1b2 + c1c2 = 0

(ii) If a line is perpendicular to the plane, at that point DR’s of line are proportional to the normal of the plane.

i.e.

where, a1, b1 and c1 are the DR’s of a line and a2, b2 and c2 are the DR’s of normal to the plane.

October 29, 2020