Introduction to 3D Geometry Class 11 Notes - Chapter 12

A line passing through the origin making angles p, q and r with x, y, 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 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].

If a, b, c are direction cosines of a line, then \(a^{2}+b^{2}+c^{2}=1\). Direction cosines of a line joining two points A(m1, n1, o1) and B(m2, n2, o2) are \(\frac{m_{2}-m_{1}}{AB} =\frac{n_{2}-n_{1}}{AB}=\frac{o_{2}-o_{1}}{AB}\)

where PQ = \(\sqrt{(m_{2}-m_{1})^{2} + (n_{2}-n_{1})^{2} +(o_{2}-o_{1})^{2} }\)

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,

\(p=\frac{m}{\sqrt{m^{2}+n^{2}+o^{2}}}\)\(q=\frac{n}{\sqrt{m^{2}+n^{2}+o^{2}}}\)\(r=\frac{o}{\sqrt{m^{2}+n^{2}+o^{2}}}\)

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

\(\frac{\left | Px + Qy + Rz + S\right |}{\sqrt{P^{2}+Q^{2}+R^{2}}}\)

Introduction to Three Dimensional Geometry Practice Questions

  1. Determine the direction cosines of a line L if it makes angles 135°, 90°, and 45° with x, y, z-axes.
  2. Show that the given points A(5, 8, 7), B (-1, -2, 1), and C(2, 3, 4) are collinear.
  3. 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.
  4. Determine the vector equation for a line L passing through (3, 4, 6) and (-1, 0, 2).
  5. Determine the cartesian and vector equations of the lines passing through center and points (5, 2, 3).
  6. Determine the Cartesian and vector equations of a line passing through points (3, – 2, 6) and (3, – 2, – 5).
  7. Determine the angle between planes 12x + 2y – 12z = 15 and 23x – 26y – 12z = 27.
  8. Find the angle between planes 13x – 26y + 12z = 17 and 12x + 12y – 12z = 8.

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Practise This Question

Let α(a) and β(a) be the roots of the equation (31+a1)+(1+a1)x+(61+a1)=0 where a>1.  then lima0+α(a)  and lima0+β(a) are