# NCERT Solutions for class 12th Mathematics Chapter 11 Three-Dimensional Geometry

#### Exercise 11.1

Question 1. If a line makes angles 90º, 135º, 45º with the x, y and z axes respectively, findits direction cosines.

Question 2. Find the direction cosines of a line which makes equal angles with coordinate axes.

Question 3. If a line has the direction ratios – 18, 12, –4 then what are its direction cosines?

Question 4. Show that the points (2, 3, 4) (–1, –2, 1), (5, 8, 7) are collinear.

Note: For three points A, B and C, if direction ratios of AB and BC are proportional, then the three points are collinear.

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).

#### Exercise 11.2

Question 1. Show that the lines with direction cosines:

Question 2. Show that the line through the points (1, –1, 2) (3, 4, –2) is perpendicular tothe line through the points (0, 3, 2) and (3, 5, 6).

Question 3. Show that the line through the points (4, 7 , 8) (2, 3, 4) is parallel to the linethrough the points (–1, –2, 1) and (1, 2, 5).

Question 4. Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector 3i+ 2j -2k.

Question 5. Find the equation of the line in vector and in cartesian form that passesthrough the point with position vector 2i -j +4k and is in the direction i +2j -k.

Question 6. Find the cartesian equation of the line which passes through the point

Question 7. The cartesian equation of a line is x-5/3 = y+4/7 = z-6/2. write its vector form.

Question 8. Find the vector and the cartesian equations of the lines that passes throughthe origin and (5, –2, 3).

Question 9. Find the vector and cartesian equations of the line that passes through thepoints (3, –2, –5), (3, –2, 6).

Question 10. Find the angle between the following pair of lines

Question 11. Find the angle between the following pair of lines

Question 12. Find the values of p so that the lines 1-x/3 = 7y-14/2p = z-3/2 and

Note: For a line to be in standard form, coefficients of variables should be positive and constant.

Question 13. Show that the lines x-5/7 = y+2/-5 and x/1 = y/2 = z/3 are perpendicular toeach other.

Question 14. Find the shortest distance between the lines

Question 15. Find the shortest distance between the lines x+1/7 = y+1/-6 = z+1/1 and x-3/1 = y-5/-2 = z-7/1.

Note: If the two lines are parallel and non-intercepting, then only we can find the shortest distance between them.

Question 16. Find the distance between the lines whose vector equations are:

Question 17. Find the shortest distance between the lines whose vector equations are

#### Exercise 11.3

Question 1. In each of the following exercises, determine the direction cosines of thenormal to the plane and the distance from the origin.

Question 2. Find the vector equation of a plane which is at a distance of 7 units from theorigin and normal to the vector 3i+ 5j – 6k.

Question 3. Find the Cartesian equation of the Following planes

Question 4. In the following cases find the coordinates of the foot of perpendicular drawn from the origin.

Question 5. Find the vector and cartesian equation of the planes

Question 6. Find the equations of the planes that passes through three points

Question 7. Find the intercepts cut off by the plane 2x + y – z = 5.

Note: Intercepts cut of by any plane are the points where the plane meets the coordinate axes.

Question 8. Find the equation of the plane with intercept 3 on the y- axis and parallel to ZOX plane.

Sol. Any plane parallel to ZOX– plane is y = b where b is the intercept on y–axis. :. b = 3.Hence equation of the required plane is y = 3

Question 9. Find the equation of the plane through the intersection of the planes 3x – y +2z –4 = 0 and x + y + z –2 = 0 and the point (2, 2, 1).

Question 10. Find the vector equation of the plane passing through the intersection of the

Question 11. Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0.

Question 12. Find the angle between the planes whose vector equations are

Question 13. In the following determine whether the given planes are parallel orperpendicular, and in case they are neither, find the angle between them.

Question 14. In the following cases, find the distance of each of the given points from the corresponding given plane.

#### Miscellaneous Exercise

Question 1. Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, –1), (4, 3, –1).

Sol. First two points A and B are (0, 0, 0) and (2, 1, 1) respectively.

:. direction ratios of AB are 2, 1, 1

Direction ratios of CD joining the points C (3, 5, –1), D (4, 3, –1) are 1, –2, 0

Now, a1a2 + b1b2 + c1c2= 2 × 1 + 1 × (–2) + 1× 0 = 2 – 2 + 0 = 0

:. AB is perpendicular to CD.

Question 2. If l1, m1, n1 and l2, m2, n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m1 n2 – m2 n1, n1 l2 – n2 l1, l1 m2 – l2 m1

Question 3. Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b.

Question 4. Find the equation of a line parallel to x-axis and passing through the origin.

Sol. The line parallel to x-axis and passing through the origin is x-axis itself.

:. equation of the line is y = 0, z = 0

Question 5. If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (–4, 3, –6) and(2, 9, 2) respectively, then find the angle betwen the lines AB and CD.

Sol. Direction ratios of AB when A and B are (1, 2, 3), (4, 5, 7) are 4 – 1, 5 – 2, 7–3 or 3, 3, 4 Direction ratios of the line joining the points C (–4, 3, –6) and D (2, 9, 2)are 2 + 4, 9 – 3, 2 + 6 or 6, 6, 8

Direction ratio of AB and CD are proportional.

=> Angle between these lines is zero. Þ The lines AB and CD are parallel.

Question 6. If the lines x-1/-3 = y-2/2k = z-3/2 and x-1/3k = y-1/1 = z-6/-5 are perpendicular, find the value of k.

Question 7. Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane r (i +2j -5k) + 9 = 0

Question 8. Find the equation of the plane passing through (a, b, c) and parallel to the plane r (i +j +k) = 2.

Question 9. Find the shortest distance between lines

Question 10. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1)crosses the YZ-plane.

Question 11. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1)crosses the ZX – plane.

Question 12. Find the coordinates of the point where the line through (3, – 4, –5) and (2, –3, 1) crosses the plane 2x + y + z = 7.

Question 13. Find the equation of the plane passing through the point (–1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.

Question 14. If the points (1, 1, p) and (–3, 0, 1) be equidistant from the planer r. (3i+ 4j -12k) + 13 = 0, then find the value of p.

Question 15. Find the equation of the plane passing through the line of intersection of the

Question 16. If O be the origin and the coordiantes of P be (1, 2, –3), then find the equation of the plane passing through P and perpendicular to OP.

Sol. The points O and P are (0, 0, 0) and (1, 2, –3)

direction ratios of OP are 1, 2, –3

The plane passing through (x1, y1, z1) is: a (x – x1) + b (y – y1) + c (z – z1) = 0 where a, b, c are the direction ratios of normal.

Direction ratios of normal are 1, 2, –3 and the point P is (1, 2, –3)

Equation of the required plane is 1 (x – 1) + 2 (y – 2) –3 (z + 3) = 0 or x + 2y – 3z – 14 = 0

Question 17. Find the equation of the plane which contains the line of intersection of the

Question 18. Find the distance of the point (–1, –5, –10) from the point of intersection of

Question 19. Find the vector equation of the line passing through (1, 2, 3) and parallel to

Question 20. Find the vector equation of the line passing through the point (1, 2, –4) and perpendicular to the two lines:

Question 21. Prove that if a plane has the intercepts a, b, c and is at a distance of p units

Choose the correct answer in Questions 22 and 23.

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