Three-Dimensional Geometry Class 12 Notes Mathematics Chapter 11

Chapter at a Glance

1. Direction Cosine of a line– Let AB be a line in space. Through origin O, draw a line OP parallel to AB. Let OP makes angles a, b, g with positive direction of x-axis, y-axis and z-axis respectively. Cosines of the angles a, b, g i.e. cos a, cos b, cos g are known as the direction cosines of line AB. Let l = cos a, m = cos b, n = cos g. l, m, n are respectively called x, y and z direction cosine of line AB. For a line, l2 + m2 + n2 = 1

2. Direction Ratio of a line– A set of three numbers which are proportional to direction cosines of a line are known as direction ratios of the line. x, y and z direction ratio of a line represented by a, b, c respectively. Relation between Direction Cosines and Direction Ratio of a Line

3. Straight Line- (Different forms of equations of a line)

4. Angles between the two lines

5. Shortest Distance between two skew lines

6. Distance Between Parallel lines-

Let the parallel lines be r= a +lamda b and

7. Coplanarity of two lines

8. Planes- (Different forms of equations of plane)

(i) The general form of the equation of the plane is Ax + By + Cz + D = 0.

(ii) (a) If a plane passes through (x1, y1, z1) and perpendicular to the line with direction ratios a, b, c then the equation of the plane is a (x – x1) + b (y – y1) + c (z – z1) = 0

(b) If a plane passes through a point with position vector a and perpendicular to vector Nuuris (r-a).N= 0

(iii) (a) Equation of the plane passing through the three non-collinear points A (x1, y1, z1), B(x2, y2, z2) and C(x3, y3, z3) is

9. The angle between the planes

11. Angle between a line and a plane– The angle between a line and a plane is said to be the complement of the angle between the line and the normal to the plane.

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