# 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.