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**Question 1.** If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its
direction cosines.

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

**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).

**Question 2. **Show that the line through the points (1, – 1, 2), (3, 4, – 2) is perpendicular to the
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 line
through the points (– 1, – 2, 1), (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 3 iˆ + 2 ˆj −2 kˆ .

**Question 5.** Find the equation of the line in vector and in cartesian form that passes through
the point with position vector 2 iˆ− j + 4 kˆ and is in the direction iˆ + 2 ˆj − kˆ .

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

**Question 10.** Find the angle between the following pairs of lines:

EXERCISE 11.3

**Question 1.** In each of the following cases, determine the direction cosines of the normal to
the plane and the distance from the origin.

(a) z = 2

(b) x + y + z = 1

(c) 2x + 3y – z = 5

(d) 5y + 8 = 0

**Question 2.** Find the vector equation of a plane which is at a distance of 7 units from the
origin and normal to the vector 3 iˆ + 5 ˆj − 6 kˆ.

**Question 3.** Find the Cartesian equation of the following planes:

(a) r (iˆ + ˆj − kˆ) = 2

(b) r (2iˆ +3 ˆj − 4kˆ) = 1

(c) r [(s − 2t) iˆ + (3 − t) ˆj +(2 s +t ) kˆ] = 15

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

(a) 2x + 3y + 4z – 12 = 0 (b) 3y + 4z – 6 = 0

(c) x + y + z = 1 (d) 5y + 8 = 0

**Question 5. **Find the vector and cartesian equations of the planes

(a) that passes through the point (1, 0, – 2) and the normal to the plane is
iˆ + ˆj − kˆ.

(b) that passes through the point (1,4, 6) and the normal vector to the plane is
iˆ−2 ˆj + kˆ.

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

(a) (1, 1, – 1), (6, 4, – 5), (– 4, – 2, 3)

(b) (1, 1, 0), (1, 2, 1), (– 2, 2, – 1)

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

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

**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
planes r .(2 iˆ + 2 ˆj − 3 kˆ ) = 7 , r .(2 iˆ + 5 ˆj + 3 kˆ ) = 9 and through the point
(2, 1, 3).

**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
r (2 iˆ + 2 ˆj − 3 kˆ) = 5 and r (3 iˆ − 3 ˆj + 5 kˆ) = 3
.

**Question 13.** In the following cases, determine whether the given planes are parallel or
perpendicular, and in case they are neither, find the angles between them.

(a) 7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0

(b) 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0

(c) 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0

(d) 2x – y + 3z – 1 = 0 and 2x – y + 3z + 3 = 0

(e) 4x + 8y + z – 8 = 0 and y + z – 4 = 0

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

Miscellaneous Exercise on Chapter 11

**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).

**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 1 2 2 1 1 2 2 1 1 2 2 1 m n − m n , n l − n l , l m − l m

**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.

**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 between the lines AB and CD.

**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 plane (3 ˆ + 4 ˆ −12 ˆ) +13 = 0,
r i j k then find the value of p.

**Question 15.** Find the equation of the plane passing through the line of intersection of the
planes r (iˆ + ˆj + kˆ) =1 and r (2 iˆ + 3 ˆj − kˆ) + 4 = 0 and parallel to x-axis.

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

**Question 17.** Find the equation of the plane which contains the line of intersection of the planes
r (iˆ + 2 ˆj + 3 kˆ) − 4 = 0 , r (2 iˆ + ˆj − kˆ) + 5 = 0 and which is perpendicular to the
plane r (5 iˆ + 3 ˆj − 6kˆ) + 8 = 0 ] .

**Question 18.** Find the distance of the point (– 1, – 5, – 10) from the point of intersection of the
line r = 2 iˆ − ˆj + 2 kˆ + λ (3 iˆ + 4 ˆj + 2 kˆ)
and the plane r (iˆ − ˆj + kˆ) = 5
.

**Question 19.** Find the vector equation of the line passing through (1, 2, 3) and parallel to the
planes r (iˆ − ˆj + 2kˆ) = 5
and r (3 iˆ + ˆj + kˆ) = 6 .

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

- Chapter 1: RELATIONS AND FUNCTIONS
**[Ques wise Ans]** - Chapter 3: MATRIX
**[Ques wise Ans]** - Chapter 4: DETERMINANTS
**[Ques wise Ans]** - Chapter 8: APPLICATION OF INTEGRALS
**[Ques wise Ans]** - Chapter 1 Relations and Functions
- Chapter 2 Inverse Trigonometric Functions
- Chapter 3 Matrices
- Chapter 4 Determinants
- Chapter 5 Continuity and Differentiability
- Chapter 6 Application of Derivatives
- Chapter 7 Integral
- Chapter 8 Application of Integrals
- Chapter 9 Differential Equations
- Chapter 10 Vector Algebra
- Chapter 11 Three Dimensional Geometry
- Chapter 12 Linear Programming
- Chapter 13 Probability

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