## CBSE 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. l^{2} + m^{2} + n^{2} = 1 or cos^{2} α + cos^{2} β + cos^{2} 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 \(\frac { l }{ a }\) = \(\frac { m }{ b }\) = \(\frac { n }{ c }\)

(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(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) are x_{2} – x_{1}, y_{2} – y_{1} and z_{2} – z_{1} 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 \(\vec { b }\)

Vector form \(\vec { r } =\vec { a } +\lambda \vec { b }\)

where, \(\vec { a }\) = Position vector of a point through which the line is passing

\(\vec { b }\) = A vector parallel to a given line

Cartesian form

where, (x_{1}, y_{1}, z_{1}) 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:** \(\vec { r } =\vec { a } +\lambda \left( \vec { b } -\vec { a } \right)\), λ ∈ R, where a and b are the position vectors of the points through which the line is passing.

**Cartesian form**

where, (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) are the points through which the line is passing.

**Angle between Two Lines**

**Vector form:** Angle between the lines \(\vec { r } =\vec { { a }_{ 1 } } +\lambda \vec { { b }_{ 1 } }\) and \(\vec { r } =\vec { { a }_{ 2 } } +\mu \vec { { b }_{ 2 } }\) is given as

**Condition of Perpendicularity:** Two lines are said to be perpendicular, when in vector form \(\vec { { b }_{ 1 } } \cdot \vec { { b }_{ 2 } } =0\); in cartesian form a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0

or l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} = 0 [direction cosine form]

**Condition that Two Lines are Parallel:** Two lines are parallel, when in vector form \(\vec { { b }_{ 1 } } \cdot \vec { { b }_{ 2 } } =\left| \vec { { b }_{ 1 } } \right| \left| \vec { { b }_{ 2 } } \right|\); in cartesian form \(\frac { { a }_{ 1 } }{ { a }_{ 2 } } =\frac { { b }_{ 1 } }{ { b }_{ 2 } } =\frac { { c }_{ 1 } }{ { c }_{ 2 } }\)

or

\(\frac { { l }_{ 1 } }{ { l }_{ 2 } } =\frac { { m }_{ 1 } }{ { m }_{ 2 } } =\frac { { n }_{ 1 } }{ { n }_{ 2 } }\)

[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 \(\vec { r } =\vec { { a }_{ 1 } } +\lambda \vec { { b }_{ 1 } }\) and \(\vec { r } =\vec { { a }_{ 2 } } +\lambda \vec { { b }_{ 2 } }\). Then, shortest distance

where \(\vec { { a }_{ 2 } }\), \(\vec { { a }_{ 1 } }\) are position vectors of point through which the line is passing and \(\vec { { b }_{ 1 } }\), \(\vec { { b }_{ 2 } }\) 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 l_{1} and l_{2} are parallel, then they are coplanar. Let the lines be \(\vec { r } =\vec { { a }_{ 1 } } +\lambda \vec { b }\) and \(\vec { r } =\vec { { a }_{ 2 } } +\mu \vec { b }\), 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 (x_{1}, y_{1}, z_{1}) and Q (x_{2}, y_{2}, z_{2}) is given by

**Mid-point of a Line:** The mid-point of a line joining points A (x_{1}, y_{1}, z_{1}) and B (x_{2}, y_{2}, z_{2}) 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 \(\vec { r } \cdot \vec { n } =d\), where \(\vec { n }\) 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 \(\vec { a }\) and perpendicular to the vector \(\vec { n }\), then \(\left( \vec { r } -\vec { a } \right) \cdot \vec { n } =0\)

This is the vector equation of the plane.

**Cartesian form:** Equation of plane passing through point (x_{1}, y_{1}, z_{1}) is given by

a (x – x_{1}) + b (y – y_{1}) + c (z – z_{1}) = 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 \(\vec { a }\), \(\vec { b }\) and \(\vec { c }\) are the position vectors of three given points, then equation of a plane passing through three non-collinear points is \(\left( \vec { r } -\vec { a } \right) \cdot \left\{ \left( \vec { b } -\vec { a } \right) \times \left( \vec { c } -\vec { a } \right) \right\} =0\).

**Cartesian form:** If (x_{1}, y_{1}, z_{1}) (x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, z_{3}) 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 \(\frac { x }{ a } +\frac { y }{ b } +\frac { z }{ c } =1\)

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

**Vector form:** If equation of the planes are \(\vec { r } \cdot \vec { { n }_{ 1 } } ={ d }_{ 1 }\) and \(\vec { r } \cdot \vec { { n }_{ 2 } } ={ d }_{ 2 }\), then equation of any plane passing through the intersection of planes is

\(\vec { r } \cdot \left( \vec { { n }_{ 1 } } +\lambda \vec { { n }_{ 2 } } \right) ={ d }_{ 1 }+\lambda { d }_{ 2 }\)

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

**Cartesian form:** If the equation of planes are a_{1}x + b_{1}y + c_{1}z = d_{1} and a_{2}x + b_{2}y + c_{2}z = d_{2}, then equation of any plane passing through the intersection of planes is a_{1}x + b_{1}y + c_{1}z – d_{1} + λ (a_{2}x + b_{2}y + c_{2}z – d_{2}) = 0

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

**Coplanarity of Two Lines**

**Vector form:** If two lines \(\vec { r } =\vec { { a }_{ 1 } } +\lambda \vec { { b }_{ 1 } }\) and \(\vec { r } =\vec { { a }_{ 2 } } +\mu \vec { { b }_{ 2 } }\) are coplanar, then

\(\left( \vec { { a }_{ 2 } } -\vec { { a }_{ 1 } } \right) \cdot \left( \vec { { b }_{ 2 } } -\vec { { b }_{ 1 } } \right) =0\)

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

**Vector form:** If \(\vec { { n }_{ 1 } }\) and \(\vec { { n }_{ 2 } }\) are normals to the planes and θ be the angle between the planes \(\vec { r } \cdot \vec { { n }_{ 1 } } ={ d }_{ 1 }\) and \(\vec { r } \cdot \vec { { n }_{ 2 } } ={ d }_{ 2 }\), 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 \(\vec { { n }_{ 1 } } \cdot \vec { { n }_{ 2 } } =0\) and parallel, if \(\vec { { n }_{ 1 } } \cdot \vec { { n }_{ 2 } } =\left| \vec { { n }_{ 1 } } \right| \left| \vec { { n }_{ 2 } } \right|\)

**Cartesian form:** If the two planes are a_{1}x + b_{1}y + c_{1}z = d_{1} and a_{2}x + b_{2}y + c_{2}z = d_{2}, then

Note: Planes are perpendicular to each other, if a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0 and planes are parallel, if \(\frac { { a }_{ 1 } }{ { a }_{ 2 } } =\frac { { b }_{ 1 } }{ { b }_{ 2 } } =\frac { { c }_{ 1 } }{ { c }_{ 2 } }\)

**Distance of a Point from a Plane**

**Vector form:** The distance of a point whose position vector is \(\vec { a }\) from the plane

\(\vec { r } \cdot \hat { n } =d\quad is\quad \left| d-\vec { a } \hat { n } \right|\)

Note:

(i) If the equation of the plane is in the form \(\vec { r } \cdot \vec { n } =d\), where \(\vec { n }\) is normal to the plane, then the perpendicular distance is \(\frac { \left| \vec { a } \cdot \vec { n } -d \right| }{ \left| \vec { n } \right| }\)

(ii) The length of the perpendicular from origin O to the plane \(\vec { r } \cdot \vec { n } =d\quad is\quad \frac { \left| d \right| }{ \left| \vec { n } \right| }\) [∵ \(\vec { a }\) = 0]

**Cartesian form:** The distance of the point (x_{1}, y_{1}, z_{1}) from the plane Ax + By + Cz = D is

**Angle between a Line and a Plane**

**Vector form:** If the equation of line is \(\vec { r } =\vec { a } +\lambda \vec { b }\) and the equation of plane is \(\vec { r } \cdot \vec { n } =d\), 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 \(\frac { a }{ l } =\frac { b }{ m } =\frac { c }{ n }\)

**Remember Points**

(i) If a line is parallel to the plane, then normal to the plane is perpendicular to the line. i.e. a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 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. \(\frac { { a }_{ 1 } }{ { a }_{ 2 } } =\frac { { b }_{ 1 } }{ { b }_{ 2 } } =\frac { { c }_{ 1 } }{ { c }_{ 2 } }\)

where, a_{1}, b_{1} and c_{1} are the DR’s of a line and a_{2}, b_{2} and c_{2} are the DR’s of normal to the plane.