Three Dimensional Geometry Class 12 Notes - Chapter 11

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 \(p^{2}+q^{2}+r^{2}=1\) and suppose we have line joining two points such as R(\(x_{1},y_{1},z_{1}\)) and S(\(x_{2},y_{2},z_{2}\)) are represented by

\(x_{2}-x_{1}/RS,y_{2}-y_{1}/RS,z_{2}-z_{1}/RS\) and RS=\(\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}\)

If a line has p,q,r are the direction cosines and a,b,c are the direction ratios then, p=\(a/\sqrt{a^{2}+b^{2}+c^{2}}\), q=\(b/\sqrt{a^{2}+b^{2}+c^{2}}\),r=\(c/\sqrt{a^{2}+b^{2}+c^{2}}\)

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 \(p_{1},q_{1},r_{1}\) and \(p_{2},q_{2},r_{2}\) are the given direction cosines and angle between two distinct lines is \(\theta\) then,

\(cos\theta\)=\(\left | p_{1}p_{2}+q_{1}q_{2}+r_{1}r_{2} \right |\)

And if two lines have direction ratios as \(l_{1},m_{1},n_{1}\) and \(l_{2},m_{2},n_{2}\) and angle \(\theta\) is given then,

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

The equation of the vector of a line

The equation of vector of the line passing through a point having position vector as \(\underset{a}{\rightarrow}\) and is parallel to a given vector \(\underset{b}{\rightarrow}\) is \(\underset{r}{\rightarrow}\)=\(\underset{a}{\rightarrow}+\lambda \underset{b}{\rightarrow}\)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 \((x_{1},y_{1},z_{1}) and (x_{2},y_{2},z_{2})\)  then \(x-x_{1}/x_{2}-x_{1}=y-y_{1}/y_{2}-y_{1}=z-z_{1}/z_{2}-z_{1}\)

Important Questions

  1. Distance between the two planes with dimensions 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is
  2. (A) 2 units (B) 4 units (C) 8 units (D) \(2/\sqrt{29}\) units

  1. 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.
  2. Find the coordinates of the point where the line through (4, – 5, – 6) and(3, – 4, 2) crosses the plane 3x + y + z = 8.
  3. Find the coordinates of the point where the line through (6, 2, 7) and (4, 5,2) crosses the YZ-plane
  4. Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b.

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Practise This Question

If |x|<1then ddx(12x+3x2........)=