A line passes through the origin making angles p, q and r with x, y, and z-axes then, the cosine of these angles, namely, cos p, cos q, and cos r are known as direction cosines of the line L. Any 3 numbers that are proportional to the direction cosines are known as the direction ratios of that line. If x, y, z are direction cosines and p, q, r are direction ratios of a line, then a = λl, b = λm, and c = λn, [where λ belongs to R].
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If a, b, and c are direction cosines of a line, then
where PQ =
The direction ratios of a line are the numbers which are proportional to the direction cosines of a line. If p, q, r are the direction cosines and m, n, o are the direction ratios of a line then,
If a1, b1, c1 and a2, b2, c2 are the direction cosines of two lines; and p is the acute angle between them;
then cosp = |a1a2 + b1b2+ c1c2 |.
The cartesian equation of a plane passing through the intersection of planes P1 x + Q1 y + R1 z + S1 = 0 and P2 x + Q2 y + R2 z + S2 = 0 is (P1 x + Q1 y + R1 z + S1) + l(P2 x + Q2 y + R2 z + S2) = 0.
The distance from a point (x, y, z) to the plane Px1 + Qy1 + Rz1 + S = 0 is
Also Refer: Three Dimensional Shapes
Introduction to Three-Dimensional Geometry Practice Questions
- Determine the direction cosines of a line L if it makes angles 135°, 90°, and 45° with x, y, and z-axes.
- Show that the given points A(5, 8, 7), B (-1, -2, 1), and C(2, 3, 4) are collinear.
- If the coordinates of the vertices of a triangle are (-1, 1, 2), (3, 5, -4), and (-5, -5, -2). Find the direction cosines of its sides.
- Determine the vector equation for a line L passing through (3, 4, 6) and (-1, 0, 2).
- Determine the cartesian and vector equations of the lines passing through the centre and points (5, 2, 3).
- Determine the Cartesian and vector equations of a line passing through points (3, – 2, 6) and (3, – 2, – 5).
- Determine the angle between planes 12x + 2y – 12z = 15 and 23x – 26y – 12z = 27.
- Find the angle between planes 13x – 26y + 12z = 17 and 12x + 12y – 12z = 8.
Related Links:
- Equation of A Plane
- Find Direction Ratios and Cosines
- Coplanarity of Two Lines In 3D Geometry
- NCERT Solutions for Class 11 Maths Chapter 12
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NCERT Exemplar for Class 11 Maths Chapter 12
Frequently Asked Questions on CBSE Class 11 Maths Notes Chapter 12 Introduction to 3D Geometry
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.
What is the dimensional plane?
In mathematics, a plane is a flat, two-dimensional surface that extends indefinitely.
What is 3D geometry?
3D geometry involves the mathematics of shapes in 3D space and involves 3 coordinates which are x-coordinate, y-coordinate and z-coordinate.
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