Chapter 11 Three Dimensional Geometry | class 12th | quick revision notes maths

Class 12 Maths Notes Chapter 11 Three Dimensional Geometry

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. Therefore, 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 direction ratios of a line, then la = mb = nc
(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 parallel lines are proportional.
(ii) Direction ratios of a 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 Line through a Given Point and parallel to a given vector b⃗ 
Vector form r⃗ =a⃗ +λb⃗ 
where, a⃗  = Position vector of a point through which the line is passing
b⃗  = 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 form: r⃗ =a⃗ +λ(b⃗ −a⃗ ), 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 r⃗ =a1→+λb1→ and r⃗ =a2→+μb2→ is given as

Condition of Perpendicularity: Two lines are said to be perpendicular, when in vector form b1→⋅b2→=0; 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 b1→⋅b2→=∣∣∣b1→∣∣∣∣∣∣b2→∣∣∣; in cartesian form a1a2=b1b2=c1c2
or
l1l2=m1m2=n1n2
[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 of the shortest distance will be perpendicular to both the lines.
Vector form: If the lines are r⃗ =a1→+λb1→ and r⃗ =a2→+λb2→. Then, shortest distance

where a2→, a1→ are position vectors of point through which the line is passing and b1→, b2→ 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 r⃗ =a1→+λb⃗  and r⃗ =a2→+μb⃗ , 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 wholly 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 r⃗ ⋅n⃗ =d, where n⃗  is a vector which is normal to the plane.
Cartesian form: The equation of the plane is given by ax + by + cz = d, where a, b and c are the direction ratios of plane and d is the distance of the plane from origin.
Another equation of the plane is lx + my + nz = p, where l, m, and n are direction cosines of the perpendicular from origin and p is a distance of a plane from origin.
Note: If d is the distance from the origin and l, m and n are the direction cosines of the normal to the plane through the origin, then the foot of the perpendicular is (ld, md, nd).

Equation of a Plane Perpendicular to a given Vector and Passing Through a given Point
Vector form: Let a plane passes through a point A with position vector a⃗  and perpendicular to the vector n⃗ , then (r⃗ −a⃗ )⋅n⃗ =0
This is the vector equation of the plane.
Cartesian form: Equation of plane passing 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 normal to the plane.

Equation of Plane Passing through Three Non-collinear Points
Vector form: If a⃗ , b⃗  and c⃗  are the position vectors of three given points, then equation of a plane passing through three non-collinear points is (r⃗ −a⃗ )⋅{(b⃗ −a⃗ )×(c⃗ −a⃗ )}=0.
Cartesian form: If (x₁, y₁, z₁) (x₂, y₂, z₂) and (x₃, y₃, z₃) are three non-collinear points, then equation of the plane is

If above points are collinear, then

Equation of Plane in Intercept Form: If a, b and c are x-intercept, y-intercept and z-intercept, respectively made by the plane on the coordinate axes, then equation of plane is xa+yb+zc=1

Equation of Plane Passing through the Line of Intersection of two given Planes
Vector form: If equation of the planes are r⃗ ⋅n1→=d1 and r⃗ ⋅n2→=d2, then equation of any plane passing through the intersection of planes is
r⃗ ⋅(n1→+λn2→)=d1+λd2
where, λ is a constant and calculated from given condition.
Cartesian form: If the equation of planes are a₁x + b₁y + c₁z = d₁ and a₂x + b₂y + c₂z = d₂, then equation of any plane passing through the intersection of planes is a₁x + b₁y + c₁z – d₁ + λ (a₂x + b₂y + c₂z – d₂) = 0
where, λ is a constant and calculated from given condition.

Coplanarity of Two Lines
Vector form: If two lines r⃗ =a1→+λb1→ and r⃗ =a2→+μb2→ are coplanar, then
(a2→−a1→)⋅(b2→−b1→)=0

Angle between Two Planes: Let θ be the angle between two planes.
Vector form: If n1→ and n2→ are normals to the planes and θ be the angle between the planes r⃗ ⋅n1→=d1 and r⃗ ⋅n2→=d2, then θ is the angle between the normals to the planes drawn from some common points.

Note: The planes are perpendicular to each other, if n1→⋅n2→=0 and parallel, if n1→⋅n2→=∣∣n1→∣∣∣∣n2→∣∣
Cartesian form: If the two planes are a₁x + b₁y + c₁z = d₁ and a₂x + b₂y + c₂z = d₂, then

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

Distance of a Point from a Plane
Vector form: The distance of a point whose position vector is a⃗  from the plane
r⃗ ⋅n^=dis|d−a⃗ n^|

Note:
(i) If the equation of the plane is in the form r⃗ ⋅n⃗ =d, where n⃗  is normal to the plane, then the perpendicular distance is ∣∣a⃗ ⋅n⃗ −d∣∣∣∣n⃗ ∣∣
(ii) The length of the perpendicular from origin O to the plane r⃗ ⋅n⃗ =dis|d|∣∣n⃗ ∣∣ [∵ a⃗  = 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 r⃗ =a⃗ +λb⃗  and the equation of plane is r⃗ ⋅n⃗ =d, then the angle θ between the line and the normal to the plane is

and so 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, then

If a line is parallel to the plane, then al + bm + cn = 0 and if line is perpendicular to the plane, then al=bm=cn

Remember Points
(i) If a line is parallel to the plane, then normal to the plane is perpendicular to the line. i.e. a₁a₂ + b₁b₂ + c₁c₂ = 0
(ii) If a line is perpendicular to the plane, then DR’s of line are proportional to the normal of the plane.
i.e. a1a2=b1b2=c1c2
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.