Mathematics · Textbook solutions

Three Dimensional Geometry

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 37 questions

11.2 Direction Cosines and Direction Ratios of a Line

10 q

Solved Examples

Worked · 5
  1. 11.1 Eg.1
    If a line makes angle 90∘90^\circ, 60∘60^\circ and 30∘30^\circ with the positive direction of xx, yy and zz-axis respectively, find its direction cosines.
  2. 11.1 Eg.2
    If a line has direction ratios 22, −1-1, −2-2, determine its direction cosines.
  3. 11.1 Eg.3
    Find the direction cosines of the line passing through the two points (−2,4,−5)(-2, 4, -5) and (1,2,3)(1, 2, 3).
  4. 11.1 Eg.4
    Find the direction cosines of xx, yy and zz-axis.
  5. 11.1 Eg.5
    Show that the points A(2,3,−4)A(2, 3, -4), B(1,−2,3)B(1, -2, 3) and C(3,8,−11)C(3, 8, -11) are collinear.

Exercise 11.1

Practice · 5
  1. Ex 11.1 Q1
    If a line makes angles 90∘90^\circ, 135∘135^\circ, 45∘45^\circ with the xx, yy and zz-axes respectively, find its direction cosines.
  2. Ex 11.1 Q2
    Find the direction cosines of a line which makes equal angles with the coordinate axes.
  3. Ex 11.1 Q3
    If a line has the direction ratios −18-18, 1212, −4-4, then what are its direction cosines?
  4. Ex 11.1 Q4
    Show that the points (2,3,4)(2, 3, 4), (−1,−2,1)(-1, -2, 1), (5,8,7)(5, 8, 7) are collinear.
  5. Ex 11.1 Q5
    Find the direction cosines of the sides of the triangle whose vertices are (3,5,−4)(3, 5, -4), (−1,1,2)(-1, 1, 2) and (−5,−5,−2)(-5, -5, -2).

11.3 Equation of a Line in Space, 11.4 Angle between Two Lines and 11.5 Shortest Distance

22 q

Solved Examples

Worked · 5
  1. 11.2 Eg.6
    Find the vector and the Cartesian equations of the line through the point (5,2,−4)(5, 2, -4) and which is parallel to the vector 3i^+2j^−8k^3\hat{i} + 2\hat{j} - 8\hat{k}.
  2. 11.2 Eg.7
    Find the angle between the pair of lines given by r⃗=3i^+2j^−4k^+λ(i^+2j^+2k^)\vec{r} = 3\hat{i} + 2\hat{j} - 4\hat{k} + \lambda(\hat{i} + 2\hat{j} + 2\hat{k}) and r⃗=5i^−2j^+μ(3i^+2j^+6k^)\vec{r} = 5\hat{i} - 2\hat{j} + \mu(3\hat{i} + 2\hat{j} + 6\hat{k}).
  3. 11.2 Eg.8
    Find the angle between the pair of lines x+33=y−15=z+34\frac{x + 3}{3} = \frac{y - 1}{5} = \frac{z + 3}{4} and x+11=y−41=z−52\frac{x + 1}{1} = \frac{y - 4}{1} = \frac{z - 5}{2}.
  4. 11.2 Eg.9
    Find the shortest distance between the lines l1l_1 and l2l_2 whose vector equations are r⃗=i^+j^+λ(2i^−j^+k^)\vec{r} = \hat{i} + \hat{j} + \lambda(2\hat{i} - \hat{j} + \hat{k}) and r⃗=2i^+j^−k^+μ(3i^−5j^+2k^)\vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu(3\hat{i} - 5\hat{j} + 2\hat{k}).
  5. 11.2 Eg.10
    Find the distance between the lines l1l_1 and l2l_2 given by r⃗=i^+2j^−4k^+λ(2i^+3j^+6k^)\vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}) and r⃗=3i^+3j^−5k^+μ(2i^+3j^+6k^)\vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu(2\hat{i} + 3\hat{j} + 6\hat{k}).

Exercise 11.2

Practice · 17
  1. Ex 11.2 Q1
    Show that the three lines with direction cosines 1213, −313, −413\frac{12}{13},\ \frac{-3}{13},\ \frac{-4}{13}; 413, 1213, 313\frac{4}{13},\ \frac{12}{13},\ \frac{3}{13}; 313, −413, 1213\frac{3}{13},\ \frac{-4}{13},\ \frac{12}{13} are mutually perpendicular.
  2. Ex 11.2 Q2
    Show that the line through the points (1,−1,2)(1, -1, 2), (3,4,−2)(3, 4, -2) is perpendicular to the line through the points (0,3,2)(0, 3, 2) and (3,5,6)(3, 5, 6).
  3. Ex 11.2 Q3
    Show that the line through the points (4,7,8)(4, 7, 8), (2,3,4)(2, 3, 4) is parallel to the line through the points (−1,−2,1)(-1, -2, 1), (1,2,5)(1, 2, 5).
  4. Ex 11.2 Q4
    Find the equation of the line which passes through the point (1,2,3)(1, 2, 3) and is parallel to the vector 3i^+2j^−2k^3\hat{i} + 2\hat{j} - 2\hat{k}.
  5. Ex 11.2 Q5
    Find the equation of the line in vector and in cartesian form that passes through the point with position vector 2i^−j^+4k^2\hat{i} - \hat{j} + 4\hat{k} and is in the direction i^+2j^−k^\hat{i} + 2\hat{j} - \hat{k}.
  6. Ex 11.2 Q6
    Find the cartesian equation of the line which passes through the point (−2,4,−5)(-2, 4, -5) and parallel to the line given by x+33=y−45=z+86\frac{x + 3}{3} = \frac{y - 4}{5} = \frac{z + 8}{6}.
  7. Ex 11.2 Q7
    The cartesian equation of a line is x−53=y+47=z−62\frac{x - 5}{3} = \frac{y + 4}{7} = \frac{z - 6}{2}. Write its vector form.
  8. Find the angle between the following pairs of lines:
    Ex 11.2 Q8(i)
    r⃗=2i^−5j^+k^+λ(3i^+2j^+6k^)\vec{r} = 2\hat{i} - 5\hat{j} + \hat{k} + \lambda(3\hat{i} + 2\hat{j} + 6\hat{k}) and r⃗=7i^−6k^+μ(i^+2j^+2k^)\vec{r} = 7\hat{i} - 6\hat{k} + \mu(\hat{i} + 2\hat{j} + 2\hat{k})
  9. Ex 11.2 Q8(ii)
    r⃗=3i^+j^−2k^+λ(i^−j^−2k^)\vec{r} = 3\hat{i} + \hat{j} - 2\hat{k} + \lambda(\hat{i} - \hat{j} - 2\hat{k}) and r⃗=2i^−j^−56k^+μ(3i^−5j^−4k^)\vec{r} = 2\hat{i} - \hat{j} - 56\hat{k} + \mu(3\hat{i} - 5\hat{j} - 4\hat{k})
  10. Find the angle between the following pair of lines:
    Ex 11.2 Q9(i)
    x−22=y−15=z+3−3\frac{x - 2}{2} = \frac{y - 1}{5} = \frac{z + 3}{-3} and x+2−1=y−48=z−54\frac{x + 2}{-1} = \frac{y - 4}{8} = \frac{z - 5}{4}
  11. Ex 11.2 Q9(ii)
    x2=y2=z1\frac{x}{2} = \frac{y}{2} = \frac{z}{1} and x−54=y−21=z−38\frac{x - 5}{4} = \frac{y - 2}{1} = \frac{z - 3}{8}
  12. Ex 11.2 Q10
    Find the values of pp so that the lines 1−x3=7y−142p=z−32\frac{1 - x}{3} = \frac{7y - 14}{2p} = \frac{z - 3}{2} and 7−7x3p=y−51=6−z5\frac{7 - 7x}{3p} = \frac{y - 5}{1} = \frac{6 - z}{5} are at right angles.
  13. Ex 11.2 Q11
    Show that the lines x−57=y+2−5=z1\frac{x - 5}{7} = \frac{y + 2}{-5} = \frac{z}{1} and x1=y2=z3\frac{x}{1} = \frac{y}{2} = \frac{z}{3} are perpendicular to each other.
  14. Ex 11.2 Q12
    Find the shortest distance between the lines r⃗=(i^+2j^+k^)+λ(i^−j^+k^)\vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) and r⃗=2i^−j^−k^+μ(2i^+j^+2k^)\vec{r} = 2\hat{i} - \hat{j} - \hat{k} + \mu(2\hat{i} + \hat{j} + 2\hat{k}).
  15. Ex 11.2 Q13
    Find the shortest distance between the lines x+17=y+1−6=z+11\frac{x + 1}{7} = \frac{y + 1}{-6} = \frac{z + 1}{1} and x−31=y−5−2=z−71\frac{x - 3}{1} = \frac{y - 5}{-2} = \frac{z - 7}{1}.
  16. Ex 11.2 Q14
    Find the shortest distance between the lines whose vector equations are r⃗=(i^+2j^+3k^)+λ(i^−3j^+2k^)\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - 3\hat{j} + 2\hat{k}) and r⃗=4i^+5j^+6k^+μ(2i^+3j^+k^)\vec{r} = 4\hat{i} + 5\hat{j} + 6\hat{k} + \mu(2\hat{i} + 3\hat{j} + \hat{k}).
  17. Ex 11.2 Q15
    Find the shortest distance between the lines whose vector equations are r⃗=(1−t)i^+(t−2)j^+(3−2t)k^\vec{r} = (1 - t)\hat{i} + (t - 2)\hat{j} + (3 - 2t)\hat{k} and r⃗=(s+1)i^+(2s−1)j^−(2s+1)k^\vec{r} = (s + 1)\hat{i} + (2s - 1)\hat{j} - (2s + 1)\hat{k}.

Miscellaneous Exercise on Chapter 11

5 q

Miscellaneous Exercise

Practice · 5
  1. Misc Q1
    Find the angle between the lines whose direction ratios are aa, bb, cc and b−cb - c, c−ac - a, a−ba - b.
  2. Misc Q2
    Find the equation of a line parallel to xx-axis and passing through the origin.
  3. Misc Q3
    If the lines x−1−3=y−22k=z−32\frac{x-1}{-3} = \frac{y-2}{2k} = \frac{z-3}{2} and x−13k=y−11=z−6−5\frac{x-1}{3k} = \frac{y-1}{1} = \frac{z-6}{-5} are perpendicular, find the value of kk.
  4. Misc Q4
    Find the shortest distance between lines r⃗=6i^+2j^+2k^+λ(i^−2j^+2k^)\vec{r} = 6\hat{i} + 2\hat{j} + 2\hat{k} + \lambda(\hat{i} - 2\hat{j} + 2\hat{k}) and r⃗=−4i^−k^+μ(3i^−2j^−2k^)\vec{r} = -4\hat{i} - \hat{k} + \mu(3\hat{i} - 2\hat{j} - 2\hat{k}).
  5. Misc Q5
    Find the vector equation of the line passing through the point (1,2,−4)(1, 2, -4) and perpendicular to the two lines: x−83=y+19−16=z−107\frac{x-8}{3} = \frac{y+19}{-16} = \frac{z-10}{7} and x−153=y−298=z−5−5\frac{x-15}{3} = \frac{y-29}{8} = \frac{z-5}{-5}.