Where Mindful Learning Begins
Where Mindful Learning Begins
This Class 12 Maths NCERT Solutions Chapter 11 Ex 11.1 page covers all 5 questions, solved step-by-step, on direction cosines and direction ratios of a line — the two ideas that anchor every equation of a line and plane you'll write for the rest of this chapter.
This exercise is a direct continuation of the direction cosines you first met in Chapter 10 (Vector Algebra), now applied purely to lines in 3D space rather than vectors. Questions 1–3 are computational: converting given angles or given direction ratios into direction cosines using l^2+m^2+n^2=1 — a relationship you'll use in almost every remaining exercise of this chapter to normalise a direction vector. Questions 4 and 5 shift to using direction ratios as a proof tool: Question 4 shows three points are collinear by checking their direction ratios are proportional (the exact 3D analogue of the vector-algebra collinearity test from Chapter 10), while Question 5 asks for the direction cosines of all three sides of a triangle in one go — good practice for keeping magnitude and sign consistent across multiple segments.
Let the direction cosines of the line be l, m, n.
l = \cos90^\circ = 0.
m = \cos135^\circ = -\dfrac{1}{\sqrt2}.
n = \cos45^\circ = \dfrac{1}{\sqrt2}.
If the line makes equal angles with the axes, its direction cosines satisfy l=m=n.
Using l^2+m^2+n^2=1: 3l^2=1 \Rightarrow l=\pm\dfrac{1}{\sqrt3}.
Magnitude = \sqrt{(-18)^2+12^2+(-4)^2} = \sqrt{324+144+16} = \sqrt{484} = 22.
Direction cosines = \left(\dfrac{-18}{22}, \dfrac{12}{22}, \dfrac{-4}{22}\right), which simplifies by dividing through by 2.
Let A(2, 3, 4), B(−1, −2, 1), C(5, 8, 7).
Direction ratios of AB: (-1-2,\,-2-3,\,1-4) = (-3,-5,-3).
Direction ratios of BC: (5-(-1),\,8-(-2),\,7-1) = (6,10,6).
Notice (6,10,6) = -2\times(-3,-5,-3), so the direction ratios of AB and BC are proportional — the two segments are parallel.
Since B is a point common to both AB and BC, and the two segments are parallel through that common point, A, B and C must lie on the same straight line.
Let A(3, 5, −4), B(−1, 1, 2), C(−5, −5, −2).
Side AB: direction ratios = (-1-3,\,1-5,\,2-(-4)) = (-4,-4,6), magnitude = \sqrt{16+16+36}=\sqrt{68}=2\sqrt{17}.
Direction cosines of AB = \left(-\dfrac{2}{\sqrt{17}}, -\dfrac{2}{\sqrt{17}}, \dfrac{3}{\sqrt{17}}\right).
Side BC: direction ratios = (-5-(-1),\,-5-1,\,-2-2) = (-4,-6,-4), magnitude = \sqrt{16+36+16}=\sqrt{68}=2\sqrt{17}.
Direction cosines of BC = \left(-\dfrac{2}{\sqrt{17}}, -\dfrac{3}{\sqrt{17}}, -\dfrac{2}{\sqrt{17}}\right).
Side CA: direction ratios = (3-(-5),\,5-(-5),\,-4-(-2)) = (8,10,-2), magnitude = \sqrt{64+100+4}=\sqrt{168}=2\sqrt{42}.
Direction cosines of CA = \left(\dfrac{4}{\sqrt{42}}, \dfrac{5}{\sqrt{42}}, -\dfrac{1}{\sqrt{42}}\right).
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