CBSE Class 12 Math Notes Chapter 11 Three Dimensional Geometry

Direction Cosine of Line CBSE Class 12 Maths Notes Chapter 11 Three Dimensional Geometry Related Links Frequently asked Questions on CBSE Class 12 Maths Notes Chapter 11 3D Geometry

Direction Cosine of Line

This is referred to the cosine of the angles subtended by a line on the positive direction of the coordinate axis. If we are given a line whose direction cosines are p,q,r, then

\(\begin{array}{l}p^{2}+q^{2}+r^{2}=1\end{array} \)

and suppose we have line joining two points such as R(

\(\begin{array}{l}x_{1},y_{1},z_{1}\end{array} \)

) and S(

\(\begin{array}{l}x_{2},y_{2},z_{2}\end{array} \)

) are represented by

\(\begin{array}{l}x_{2}-x_{1}/RS,y_{2}-y_{1}/RS,z_{2}-z_{1}/RS\end{array} \)

and

RS=

\(\begin{array}{l}\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}\end{array} \)

If a line has p,q,r are the direction cosines and a,b,c are the direction ratios then,

p=

\(\begin{array}{l}a/\sqrt{a^{2}+b^{2}+c^{2}}\end{array} \)

,

q=

\(\begin{array}{l}b/\sqrt{a^{2}+b^{2}+c^{2}}\end{array} \)

,

r=

\(\begin{array}{l}c/\sqrt{a^{2}+b^{2}+c^{2}}\end{array} \)

For More Information On Direction Cosine Of A Line, Watch The Below Video.

What are Skew Lines?

These are referred to the lines in space which neither intersecting nor parallel. They exist in a different plane. The angle between two lines which intersect each other can be found from any point which is parallel to each of the skew lines.

If

\(\begin{array}{l}p_{1},q_{1},r_{1}\end{array} \)

and

\(\begin{array}{l}p_{2},q_{2},r_{2}\end{array} \)

are the given direction cosines and angle between two distinct lines is

\(\begin{array}{l}\theta\end{array} \)

then,

\(\begin{array}{l}cos\theta\end{array} \)

=

\(\begin{array}{l}\left | p_{1}p_{2}+q_{1}q_{2}+r_{1}r_{2} \right |\end{array} \)

And if two lines have direction ratios as

\(\begin{array}{l}l_{1},m_{1},n_{1}\end{array} \)

and

\(\begin{array}{l}l_{2},m_{2},n_{2}\end{array} \)

and angle

\(\begin{array}{l}\theta\end{array} \)

is given then,

\(\begin{array}{l}\left | l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2} \right/\sqrt{l_{1}^{2}+m_{1}^{2}+n_{1}^{2} \sqrt{l_{2}^{2}+m_{2}^{2}+n_{2}^{2}}}\end{array} \)

The equation of the vector of a line

The equation of vector of the line passing through a point having position vector as

\(\begin{array}{l}\underset{a}{\rightarrow}\end{array} \)

and is parallel to a given vector

\(\begin{array}{l}\underset{b}{\rightarrow}\end{array} \)

is

\(\begin{array}{l}\underset{r}{\rightarrow}\end{array} \)

=

\(\begin{array}{l}\underset{a}{\rightarrow}+\lambda \underset{b}{\rightarrow}\end{array} \)

Equation of a plane which is at a distance of d from the origin and the direction
cosines of the normal to the plane as l, m, n is lx + my + nz = d and Shortest distance between two skew lines is the line segment perpendicular to both the lines.

Cartesian Equation of a Line passing through two points

If there are two points with dimensions

\(\begin{array}{l}(x_{1},y_{1},z_{1}) and (x_{2},y_{2},z_{2})\end{array} \)

then

\(\begin{array}{l}x-x_{1}/x_{2}-x_{1}=y-y_{1}/y_{2}-y_{1}=z-z_{1}/z_{2}-z_{1}\end{array} \)

Also Access

NCERT Solutions for Class 12 Maths Chapter 11

NCERT Exemplar for Class 12 Maths Chapter 11

Important Questions

1. Distance between the two planes with dimensions 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is

(A) 2 units

(B) 4 units

(C) 8 units

(D)

\(\begin{array}{l}2/\sqrt{29}\end{array} \)

units

2. If P be the origin and the coordinates of P be (2, 3, – 4), then find the equation of the plane passing through Q and perpendicular to PQ.

3. Find the coordinates of the point where the line through (4, – 5, – 6) and (3, – 4, 2) crosses the plane 3x + y + z = 8.

4. Find the coordinates of the point where the line through (6, 2, 7) and (4, 5,2) crosses the YZ-plane

5. Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b.

Also Read:

Equation Of A Line

Equation Of A Line In Three Dimensions

Frequently asked Questions on CBSE Class 12 Maths Notes Chapter 11: 3D Geometry

Q1

What is a ‘Coordinate’ point?

The coordinates of a point are a pair of numbers that define its exact location on a two-dimensional plane.

Q2

What is ‘Dimensional plane’?

In mathematics, a plane is a flat, two-dimensional surface that extends indefinitely.

Q3

What is ‘3D geometry’?

3D geometry involves the mathematics of shapes in 3D space and involving 3 coordinates which are x-coordinate, y-coordinate and z-coordinate.

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