NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1 Three Dimensional Geometry - PDF Download

1. Direction Cosines and Direction Ratios of a Line

In the previous chapter, You have learnt that if a directed line L passing through the origin makes angles α, β and γ with x, y and z-axes, respectively, called direction angles, then cosine of these angles, namely, cos α, cos β and cos γ are called direction cosines of the directed line L.

If we reverse the direction of L, then the direction angles are replaced by their supplements, i.e., -, - and - . Thus, the signs of the direction cosines are reversed.

Remember that a given line in space can be extended in two opposite directions and that is why it has two sets of direction cosines. In order to have a unique set of direction cosines for a given line in space, you must take the given line as a directed line. The unique direction cosines are denoted by l, m and n. 

If l, m, n are the direction cosines of a line, then l2 + m2 + n2 = 1.

Direction Ratio - Any three numbers which are proportional to the direction cosines of a line are called the direction ratios of the line.  If l, m, n are direction cosines and a, b, c are direction ratios of a line, then a = λl, b=λm and c = λn, for any nonzero λ ∈ R.

If l, m, n are the direction cosines and a, b, c are the direction ratios of a line then 

$l=\frac{a}{\sqrt{a^2+b^2+c^2}} ; m=\frac{b}{\sqrt{a^2+b^2+c^2}} ; n=\frac{c}{\sqrt{a^2+b^2+c^2}}$

• Direction cosines of a line passing through two points

Direction cosines of a line joining two points P(x1 , y1 , z1 ) and Q(x2 , y2 , z2 ) are 

                    $\frac{x_2-x_1}{\mathrm{PQ}}, \frac{y_2-y_1}{\mathrm{PQ}}, \frac{z_2-z_1}{\mathrm{PQ}}$

where PQ = $\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}$

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Frequently Asked Questions

Question 1. What is the direction cosines of a line?

Answer 1. Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axes.

Question 2. What are the direction ratios of a line?

Answer 2. Direction ratios of a line are the numbers which are proportional to the direction cosines of a line.