Chapter 27 Direction Cosines and Direction Ratios Exercise Ex. 27.1

Question 1

If a line makes angles of 90°, 60° and 30° with the positive direction of x,y and z-axis respectively, find its direction cosines.Solution 1

Let l, m and n be the direction cosines of a line.

l = cos 90° = 0

begin mathsize 12px style text m end text equals text cos   60 end text degree equals 1 half end style
begin mathsize 12px style straight n equals cos text   end text 30 degree equals fraction numerator square root of 3 over denominator 2 end fraction end style
begin mathsize 12px style therefore straight T text he   direction   cosines   of   the   line   are   0 , end text 1 half comma fraction numerator square root of 3 over denominator 2 end fraction. end style

Question 2

If a line has direction ratios 2, -1, -2, determine its direction cosines.Solution 2

begin mathsize 12px style table attributes columnalign left end attributes row cell Let text    end text the text    end text direction text    end text cosines text    end text of text    end text the text    end text line text    end text be text    end text straight l comma straight m comma straight n. end cell row cell Here comma end cell row cell straight a equals 2 comma straight b equals negative 1 comma straight c equals negative 2 text    end text are text    end text the text    end text direction text    end text ratios text    end text of text    end text the text    end text line. end cell row cell straight l equals plus-or-minus fraction numerator straight a over denominator square root of straight a squared plus straight b squared plus straight c squared end root end fraction comma straight m equals plus-or-minus fraction numerator straight b over denominator square root of straight a squared plus straight b squared plus straight c squared end root end fraction comma straight n equals plus-or-minus fraction numerator straight c over denominator square root of straight a squared plus straight b squared plus straight c squared end root end fraction end cell row cell straight l equals fraction numerator 2 over denominator square root of 2 squared plus left parenthesis negative 1 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared end root end fraction comma straight m equals fraction numerator negative 1 over denominator square root of 2 squared plus left parenthesis negative 1 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared end root end fraction comma straight n equals fraction numerator negative 2 over denominator square root of 2 squared plus left parenthesis negative 1 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared end root end fraction end cell row cell straight l equals fraction numerator 2 over denominator square root of 9 end fraction comma straight m equals fraction numerator negative 1 over denominator square root of 9 end fraction comma straight n equals fraction numerator negative 2 over denominator square root of 9 end fraction end cell row cell straight l equals 2 over 3 comma straight m equals negative 1 third comma straight n equals negative 2 over 3 end cell row cell therefore The text    end text direction text    end text ratios text    end text of text    end text the text    end text line text    end text are text    end text 2 over 3 comma negative 1 third comma negative 2 over 3. end cell end table end style

Question 3

Find the direction cosines of the line passing through two points (-2, 4, -5) and (1, 2, 3).Solution 3

begin mathsize 12px style table attributes columnalign left end attributes row cell The text    end text direction text    end text ratios text    end text of text    end text the text    end text line text    end text joining text    end text left parenthesis negative 2 comma 4 comma negative 5 right parenthesis text   end text and text   end text left parenthesis 1 comma text   end text 2 comma text   end text 3 right parenthesis text   are end text comma end cell row cell left parenthesis 1 plus 2 comma 2 minus 4 comma 3 plus 5 right parenthesis equals left parenthesis 3 comma negative 2 comma 8 right parenthesis end cell row cell Here comma straight a equals 3 comma straight b equals negative 2 comma straight c equals 8 end cell row cell Direction text    end text cosines text    end text are end cell row cell fraction numerator 3 over denominator square root of 3 squared plus left parenthesis negative 2 right parenthesis squared plus 8 squared end root end fraction comma fraction numerator negative 2 over denominator square root of 3 squared plus left parenthesis negative 2 right parenthesis squared plus 8 squared end root end fraction comma fraction numerator 8 over denominator square root of 3 squared plus left parenthesis negative 2 right parenthesis squared plus 8 squared end root end fraction end cell row cell equals fraction numerator 3 over denominator square root of 77 end fraction comma fraction numerator negative 2 over denominator square root of 77 end fraction comma fraction numerator 8 over denominator square root of 77 end fraction end cell end table end style

Question 4

Using direction ratios show that the points A (2, 3, -4), (1, -2, 3) and (3, 8, -11) are collinear.Solution 4

begin mathsize 12px style table attributes columnalign left end attributes row cell Here text    end text straight A text   end text left parenthesis 2 comma 3 comma negative text 4 end text right parenthesis comma text   end text straight B text   end text left parenthesis 1 comma negative 2 comma 3 right parenthesis text   end text and text   end text straight C text   end text left parenthesis 3 comma 8 comma negative 11 right parenthesis. end cell row cell Direction text    end text ratios text    end text of text    end text AB equals left parenthesis 1 minus 2 comma negative 2 minus 3 comma 3 plus 4 right parenthesis equals left parenthesis negative 1 comma negative 5 comma 7 right parenthesis end cell row cell Direction text    end text ratios text    end text of text    end text BC equals left parenthesis 3 minus 1 comma 8 plus 2 comma negative 11 minus 3 right parenthesis equals left parenthesis 2 comma 10 comma negative 14 right parenthesis end cell row blank row cell Here comma text    end text the text    end text respective text    end text direction text    end text cosines text    end text of text    end text AB text    end text and text    end text AC comma end cell row cell fraction numerator negative 1 over denominator 2 end fraction equals fraction numerator negative 5 over denominator 10 end fraction equals fraction numerator 7 over denominator negative 14 end fraction text    end text are text    end text proportional. end cell row blank row cell Also comma text   end text straight B text    end text is text    end text the text    end text common text    end text point text    end text between text    end text the text    end text two text    end text lines comma end cell row cell therefore The text    end text points text     end text straight A text   end text left parenthesis 2 comma 3 comma negative text 4 end text right parenthesis comma text   end text straight B text   end text left parenthesis 1 comma negative 2 comma 3 right parenthesis text   end text and text   end text straight C text   end text left parenthesis 3 comma 8 comma negative 11 right parenthesis text    end text are text    end text collinear. text   end text end cell row blank end table end style

Question 5

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2)Solution 5

begin mathsize 12px style table attributes columnalign left end attributes row cell straight A left parenthesis 3 comma 5 comma negative 4 right parenthesis comma straight B left parenthesis negative 1 comma 1 comma 2 right parenthesis text    end text and text    end text straight C left parenthesis negative 5 comma negative 5 comma negative 2 right parenthesis end cell row cell The text    end text direction text    end text ratios text    end text of text    end text the text    end text side text    end text AB equals left parenthesis negative 1 minus 3 comma 1 minus 5 comma 2 plus 4 right parenthesis end cell row cell equals left parenthesis negative 4 comma negative 4 comma 6 right parenthesis end cell row cell Direction text    end text cosines text    end text of text    end text AB text    end text will text    end text be end cell row cell fraction numerator negative 4 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared plus 6 squared end root end fraction comma fraction numerator negative 4 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared plus 6 squared end root end fraction comma fraction numerator 6 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared plus 6 squared end root end fraction end cell row cell equals fraction numerator negative 4 over denominator square root of 68 end fraction comma fraction numerator negative 4 over denominator square root of 68 end fraction comma fraction numerator 6 over denominator square root of 68 end fraction end cell row cell equals fraction numerator negative 2 over denominator square root of 17 end fraction comma fraction numerator negative 2 over denominator square root of 17 end fraction comma fraction numerator 3 over denominator square root of 17 end fraction end cell row cell The text    end text direction text    end text ratios text    end text of text    end text the text    end text side text    end text BC equals left parenthesis negative 5 plus 1 comma negative 5 minus 1 comma negative 2 minus 2 right parenthesis end cell row cell equals left parenthesis negative 4 comma negative 6 comma negative 4 right parenthesis end cell row cell Direction text    end text cosines text    end text of text    end text BC text    end text will text    end text be end cell row cell fraction numerator negative 4 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 6 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root end fraction comma fraction numerator negative 6 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 6 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root end fraction comma fraction numerator negative 4 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 6 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root end fraction end cell row cell equals fraction numerator negative 4 over denominator square root of 68 end fraction comma fraction numerator negative 6 over denominator square root of 68 end fraction comma fraction numerator negative 4 over denominator square root of 68 end fraction end cell row cell equals fraction numerator negative 2 over denominator square root of 17 end fraction comma fraction numerator negative 3 over denominator square root of 17 end fraction comma fraction numerator negative 2 over denominator square root of 17 end fraction end cell row cell The text    end text direction text    end text ratios text    end text of text    end text the text    end text side text    end text AC equals left parenthesis negative 5 minus 3 comma negative 5 minus 5 comma negative 2 plus 4 right parenthesis end cell row cell equals left parenthesis negative 8 comma negative 10 comma 2 right parenthesis end cell row cell Direction text    end text cosines text    end text of text    end text AC text    end text will text    end text be end cell row cell fraction numerator negative 8 over denominator square root of left parenthesis negative 8 right parenthesis squared plus left parenthesis negative 10 right parenthesis squared plus 2 squared end root end fraction comma fraction numerator negative 10 over denominator square root of left parenthesis negative 8 right parenthesis squared plus left parenthesis negative 10 right parenthesis squared plus 2 squared end root end fraction comma fraction numerator 2 over denominator square root of left parenthesis negative 8 right parenthesis squared plus left parenthesis negative 10 right parenthesis squared plus 2 squared end root end fraction end cell row cell equals fraction numerator negative 8 over denominator square root of 168 end fraction comma fraction numerator negative 10 over denominator square root of 168 end fraction comma fraction numerator 2 over denominator square root of 168 end fraction end cell row cell equals fraction numerator negative 4 over denominator square root of 42 end fraction comma fraction numerator negative 5 over denominator square root of 42 end fraction comma fraction numerator 1 over denominator square root of 42 end fraction end cell end table end style

Question 6

Solution 6

Question 7

Solution 7

Question 8

Find the acute angle between the lines whose direction ratios are proportional to 2:3:6 and 1:2:2.Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16(i)

Solution 16(i)

Question 16(ii)

Solution 16(ii)

Question 16(iii)

Solution 16(iii)

Question 16(iv)

Find the angle between the lines whose direction cosines are given by equations

2l + 2m – n = 0, mn + ln + lm = 0Solution 16(iv)


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