# NCERT Solutions Class 12 Maths Chapter-11 (Three Dimensional Geometry)Exercise 11.1

NCERT Solutions Class 12 Maths from class 12th Students will get the answers of Chapter-11 (Three Dimensional Geometry)Exercise 11.1 This chapter will help you to learn the basics and you should expect at least one question in your exam from this chapter.
We have given the answers of all the questions of NCERT Board Mathematics Textbook in very easy language, which will be very easy for the students to understand and remember so that you can pass with good marks in your examination.

Class 12 Mathematics

Chapter-11 (Three Dimensional Geometry)

Questions and answers given in practice

Chapter-11 (Three Dimensional Geometry)

### Exercise 11.1

Q1. If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its direction cosines.
Answer.  Therefore, the direction cosines of the line are .
Q2. Find the direction cosines of a line which makes equal angles with the coordinate axes.
Answer. Let the direction cosines of the line make an angle a with each of the coordinate axes. $\therefore l=\mathrm{cos}a,m=\mathrm{cos}a,n=\mathrm{cos}$ $\begin{array}{l}{l}^{2}+{m}^{2}+{n}^{2}=1\\ ⇒{\mathrm{cos}}^{2}\alpha +{\mathrm{cos}}^{2}\alpha +{\mathrm{cos}}^{2}\alpha =1\\ ⇒3{\mathrm{cos}}^{2}\alpha =1\\ ⇒{\mathrm{cos}}^{2}\alpha =\frac{1}{3}\\ ⇒\mathrm{cos}\alpha =±\frac{1}{\sqrt{3}}\end{array}$ Thus, the direction cosines of the line, which is equally inclined to the coordinate axes, are .
Q3. If a line has the direction ratios –18, 12, – 4, then what are its direction cosines ?

Answer. If a line has the direction ratios of -18, 12, and -4 then its direction cosines are  Thus, the direction cosines are .

Q4. Show that the points (2, 3, 4), (– 1, – 2, 1), (5, 8, 7) are collinear.
Answer. The vertices of  The direction ratios of sides AB are   Therefore, the direction cosines of AB are $\frac{-4}{\sqrt{\left(-4{\right)}^{2}+\left(-4{\right)}^{2}+\left(6{\right)}^{2}}}\cdot \frac{-4}{\sqrt{\left(-4{\right)}^{2}+\left(-4{\right)}^{2}+\left(6{\right)}^{2}}},\frac{6}{\sqrt{\left(-4{\right)}^{2}+\left(-4{\right)}^{2}+\left(6{\right)}^{2}}}$ $\begin{array}{l}\frac{-4}{2\sqrt{17}},-\frac{4}{2\sqrt{17}}\cdot \frac{6}{2\sqrt{17}}\\ \frac{-2}{\sqrt{17}}\cdot \frac{-2}{\sqrt{17}},\frac{3}{\sqrt{17}}\end{array}$  $\begin{array}{l}\frac{-4}{\sqrt{\left(-4{\right)}^{2}+\left(-6{\right)}^{2}+\left(-4{\right)}^{2}}}+\frac{-6}{\sqrt{\left(-4{\right)}^{2}+\left(-6{\right)}^{2}+\left(-4{\right)}^{2}}},\frac{-4}{\sqrt{\left(-4{\right)}^{2}+\left(-6{\right)}^{2}+\left(-4{\right)}^{2}}}\\ \frac{-4}{2\sqrt{17}}\cdot \frac{-6}{2\sqrt{17}}\cdot \frac{-4}{2\sqrt{17}}\end{array}$ The direction ratios of CA are  Therefore, the direction cosines of AC are