Vector Algebra(Class-XII)
Section-I(each question carries 1 mark)
1. If |𝑎⃗| = 3 and −4 ≤ 𝑘 ≤ 1, then what can you say about |𝑘𝑎⃗| ?
⃗⃗⃗⃗⃗⃗.
2. If P (-1, 2, 5) and Q (3, 0, -2), find the unit vector in the direction of 𝑃𝑄
3. Find the values of x for which 𝑥(𝑖̂ + 𝑗̂ + 𝑘̂) is a unit vector?
4. Find the vector in the direction of the vector −𝑖̂ + 2𝑗̂ + 2𝑘̂ that has
magnitude 7.
5. If 𝑎⃗ = 𝑖̂ + 𝑗̂ + 𝑘̂, 𝑏⃗⃗ = 2𝑖̂ − 𝑗̂ + 3𝑘̂ 𝑎𝑛𝑑 𝑐⃗ = 𝑖̂ − 2𝑗̂ + 𝑘̂, find a unit vector
parallel to the vector 2𝑎⃗ − 𝑏⃗⃗ + 3𝑐⃗.
6. If 𝑎⃗ is a unit vector and (2𝑎⃗ + 𝑏⃗⃗). (2𝑎⃗ − 𝑏⃗⃗) = 2, then find |𝑏⃗⃗|.
7. If (𝑎⃗ + 𝑏⃗⃗). (𝑎⃗ − 𝑏⃗⃗) = 8 and |𝑎⃗| =8|𝑏⃗⃗|, 𝑓𝑖𝑛𝑑 |𝑎⃗| and |𝑏⃗⃗|.
8. If the angle between two vectors 𝑎⃗ 𝑎𝑛𝑑 𝑏⃗⃗ of equal magnitude is 300 and their
scalar product is 2√3, find their magnitudes.
9. Find the angle between two vectors 𝑎⃗ 𝑎𝑛𝑑 𝑏⃗⃗ with magnitudes √3 and 2
respectively and 𝑎⃗ . 𝑏⃗⃗ = √6 .
10. If 𝑎⃗ = 𝑖̂ + 2𝑗̂ − 3𝑘̂ and 𝑏⃗⃗ =3𝑖̂ − 𝑗̂ + 2𝑘̂, show that the vectors 𝑎⃗ + 𝑏⃗⃗ and 𝑎⃗ −
𝑏⃗⃗ are perpendicular to each other.
11. Find the value of 𝜆 so that the two vectors 2𝑖̂ + 3𝑗̂ − 𝑘̂ and −4𝑖̂ − 6𝑗̂ + 𝜆𝑘̂
are parallel to each other.
12. Find the value of 𝜆 so that the two vectors 2𝑖̂ + 3𝑗̂ − 𝑘̂ and −4𝑖̂ − 6𝑗̂ + 𝜆𝑘̂
are perpendicular to each other.
13. If 𝑎⃗ = 𝑖̂ + 2𝑗̂ + 9𝑘̂ and 𝑏⃗⃗ = 𝑖̂ + 𝜆𝑗̂ + 3𝑘̂, find the value of 𝜆 so that 𝑎⃗ + 𝑏⃗⃗ is
perpendicular to 𝑎⃗ − 𝑏⃗⃗ .
13. Let 𝑎⃗ = 𝑖̂ + 3𝑗̂ + 7𝑘̂ and 𝑏⃗⃗ = 7𝑖̂ − 𝑗̂ + 8𝑘̂, find the projection of 𝑎⃗𝑜𝑛𝑏⃗⃗ .
14. Let 𝑎⃗ = 𝑖̂ + 3𝑗̂ + 7𝑘̂ and 𝑏⃗⃗ = 7𝑖̂ − 𝑗̂ + 8𝑘̂, find the projection of 𝑏⃗⃗𝑜𝑛 𝑎⃗.
15. Find 𝜆 when the scalar projection of 𝑎⃗ = 𝜆𝑖̂ + 𝑗̂ + 4𝑘̂ on 𝑏⃗⃗ = 2𝑖̂ + 6𝑗̂ + 3𝑘̂ is
4 units.
𝑎⃗ = 3𝑖̂ + 𝑥𝑗̂ − 𝑘̂ and 𝑏⃗⃗ = 2𝑖̂ + 𝑗̂ + 𝑦𝑘̂
perpendicular. Given that |𝑎⃗| = |𝑏⃗⃗|, find the values of x and y.
16.
The
vectors
are
mutually
17. The scalar product of the vector 𝑖̂ + 𝑗̂ + 𝑘̂ with a unit vector along the sum of
the vectors 2𝑖̂ + 4𝑗̂ − 5𝑘̂ and 𝜆𝑖̂ + 2𝑗̂ + 3𝑘̂ is equal to one. Find the value of 𝜆.
18. If two vectors 𝑎⃗𝑎𝑛𝑑𝑏⃗⃗ are such that |𝑎⃗| = 2, |𝑏⃗⃗|=3 and 𝑎⃗. 𝑏⃗⃗=4, find |𝑎⃗ − 𝑏⃗⃗|.
19. If the sum of two unit vectors is a unit vector, show that magnitude of their
difference is √3.
20. If 𝑎⃗𝑎𝑛𝑑𝑏⃗⃗ are unit vectors and 𝜃 is the angle between them, then show that
𝜃 1
sin = |𝑎⃗ − 𝑏⃗⃗|.
2
2
2
21. Prove Cauchy-Schwartz’s inequality (𝑎⃗. 𝑏⃗⃗) ≤ |𝑎⃗|2 |𝑏|2 .
22. If 𝑎⃗ = 2𝑖̂ − 3𝑗̂, 𝑏⃗⃗ = 𝑖̂ + 𝑗̂ − 𝑘̂ 𝑎𝑛𝑑 𝑐⃗ = 3𝑖̂ − 𝑘̂, 𝑓𝑖𝑛𝑑 [𝑎⃗
𝑏⃗⃗
𝑐⃗].
23. Find 𝜆 such that the vectors 2𝑖̂ − 𝑗̂ + 𝑘̂, 𝑖̂ + 2𝑗̂ − 3𝑘̂ and 3𝑖̂ + 𝜆𝑗̂ + 5𝑘̂ are
coplanar.
24. Show that 𝑎⃗. (𝑎⃗ × 𝑏⃗⃗) = 0 for any pair 𝑎⃗, 𝑏⃗⃗ of vectors.
SECTION-B(each question carries 4 marks)
1. Prove, using vectors: If the diagonals of a Parallelogram are equal in length,
then it is a rectangle.
2. Prove, using vectors: The diagonals of a rhombus are perpendicular bisectors
of each other.
3. Prove by vector method that an angle subtended at the circumference of a
circle by any diameter is a right angle.
4. In a triangle OAB, ∠𝐴𝑂𝐵 = 900 . If P and Q are the points of trisection of AB,
5
prove that 𝑂𝑃2 + 𝑂𝑄2 = 𝐴𝐵2 .
9
5. Prove by vector method that in any triangle ABC, 𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶.
OR
Prove the cosine law by vector method.
6. Prove the Projection formula by vector method.
OR
Prove by vector method that in any triangle ABC, 𝑐 = 𝑎 cos 𝐵 + 𝑏 cos 𝐴.
7. Prove by vector method: cos(𝛼 − 𝛽) = cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽 .
8. Prove by vector method that the altitudes of a triangle are concurrent.
9. If |𝑎⃗ + 𝑏⃗⃗| = |𝑎⃗ − 𝑏⃗⃗|, where 𝑎⃗𝑎𝑛𝑑𝑏⃗⃗ are non-zero vectors, then show that
𝑎⃗𝑎𝑛𝑑𝑏⃗⃗ are perpendicular vectors.
10. If 𝑎⃗, 𝑏⃗⃗ are two vectors such that |𝑎⃗ + 𝑏⃗⃗| = |𝑎⃗|, then prove that 2𝑎⃗ + 𝑏⃗⃗ is
perpendicular to 𝑏⃗⃗ .
11. Show that the points A,B,C whose position vectors are 𝑎⃗ = 2𝑖̂ − 𝑗̂ + 𝑘̂, 𝑏⃗⃗ =
𝑖̂ − 3𝑗̂ − 5𝑘̂ 𝑎𝑛𝑑 𝑐⃗ = 3𝑖̂ − 4𝑗̂ − 4𝑘̂ respectively are the vertices of a right angled
triangle. Also find the remaining angles of the triangle.
12. Prove the Sine formula for a triangle by vector method.
13. Prove by vector method: sin(𝛼 + 𝛽) = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽 .
14. If 𝑎⃗, 𝑏⃗⃗, 𝑐⃗ are vectors such that 𝑎⃗. 𝑏⃗⃗ = 𝑎⃗. 𝑐⃗; 𝑎⃗ × 𝑏⃗⃗ = 𝑎⃗ × 𝑐⃗; 𝑎⃗ ≠ ⃗0⃗
prove that 𝑏⃗⃗ = 𝑐⃗.
2
15. show that[𝑏⃗⃗ × 𝑐⃗, 𝑐⃗ × 𝑎⃗, 𝑎⃗ × 𝑏⃗⃗] = [𝑎⃗ 𝑏⃗⃗ 𝑐⃗] .
16. Find the moment of the couple formed by the forces5𝑖̂ + 𝑘̂ and −5𝑖̂ − 𝑘̂
acting at the points (9, -1, 2) and 3, -2, 1) respectively.
17. Find the moment about the point 𝑖̂ + 2𝑗̂ − 𝑘̂ of a force represented by 𝑖̂ +
2𝑗̂ + 𝑘̂ acting through the point 2𝑖̂ + 3𝑗̂ + 𝑘̂.
18. Prove using vectors that if a,b,c are the lengths of the sides of a triangle, its
1
area is ∆= √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) where (𝑎 + 𝑏 + 𝑐).
2
⃗⃗ = −𝑖̂ + 2𝑗̂ + 𝑘̂ and 𝐶⃗ = 3𝑖̂ + 𝑗̂. Find a unit vector
19. Given: 𝐴⃗ = 𝑖̂ + 2𝑗̂ + 3𝑘̂, 𝐵
⃗⃗ which is
in the direction of the resultant of these vectors. Also find a vector 𝐷
⃗⃗. What is the inclination of 𝐷
⃗⃗ and 𝐶⃗?
normal to both 𝐴⃗ 𝑎𝑛𝑑 𝐵
20. Show that the vector area of the triangle whose vertices have position
1
vectors 𝑎⃗, 𝑏⃗⃗, 𝑐⃗ is (𝑏⃗⃗ × 𝑐⃗ + 𝑐⃗ × 𝑎⃗ + 𝑎⃗ × 𝑏⃗⃗).
2
21. If 𝑎⃗ = 4𝑖̂ + 2𝑗̂ − 𝑘̂ 𝑎𝑛𝑑 𝑏⃗⃗ = 5𝑖̂ + 2𝑗̂ − 3𝑘̂, find the angle between the
vectors 𝑎⃗ + 𝑏⃗⃗ and 𝑎⃗ − 𝑏⃗⃗.
22. If 𝑎⃗ makes equal angles with the coordinate axes and has magnitude 3, then
find the angle between 𝑎⃗ and each of the three coordinate axes.
23. If a unit vector 𝑎⃗ makes angles
𝜋
4
𝑎𝑛𝑑
𝜋
3
with x-axis and y-axis respectively
and an acute angle 𝜃 with z-axis, then find 𝜃 and the (scalar and vector)
components of 𝑎⃗ along the axes.
24. Find a vector whose magnitude is 3 units and which is perpendicular to the
vectors 𝑎⃗ and 𝑏⃗⃗ where 𝑎⃗ = 3𝑖̂ + 𝑗̂ − 4𝑘̂ 𝑎𝑛𝑑 𝑏⃗⃗ = 6𝑖̂ + 5𝑗̂ − 2𝑘̂.
24. Show that the vectors 2𝑖̂ − 𝑗̂ + 𝑘̂, 𝑖̂ − 3𝑗̂ − 5𝑘̂ 𝑎𝑛𝑑 3𝑖̂ − 4𝑗̂ − 4𝑘̂ from the
vertices of a right triangle.
25. If the vertices A,B,C of a triangle ABC have position vectors (1,2,3), (-1,0,0),
(0,1,2) respectively, then find ∠𝐴𝐵𝐶.
26. Three vertices of a triangle are A(0,-1,-2), B(3,1,4) and C(5,7,1). Show that it
is a right angle triangle. Also find the other two angles.
⃗⃗⃗⃗⃗2 where ⃗⃗⃗⃗
27. Given: 𝑎⃗ = 3𝑖̂ − 𝑗̂ 𝑎𝑛𝑑 𝑏⃗⃗ = 2𝑖̂ + 𝑗̂ − 3𝑘̂, express 𝑏⃗⃗ as ⃗⃗⃗⃗
𝑏1 +𝑏
𝑏1 is
parallel to 𝑎⃗ and ⃗⃗⃗⃗⃗
𝑏2 is perpendicular to 𝑎⃗.
28. Let 𝑎⃗, 𝑏⃗⃗ 𝑎𝑛𝑑 𝑐⃗ be three vectors of magnitudes 3, 4 and 5 units respectively. If
each of these is perpendicular to the sum of the other two vectors, find
|𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗|.
⃗⃗. Evaluate the
29. Three vectors 𝑎⃗, 𝑏⃗⃗ 𝑎𝑛𝑑 𝑐⃗ satisfy the condition 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ =0
quantity 𝜇 = 𝑎⃗. 𝑏⃗⃗ + 𝑏⃗⃗. 𝑐⃗ + 𝑐⃗. 𝑎⃗ if |𝑎⃗| = 1, |𝑏⃗⃗| = 4 𝑎𝑛𝑑 |𝑐⃗| = 2.
30. Find 𝜆 if the volume of the parallelepiped whose edges are represented by
−12𝑖̂ + 𝜆𝑘̂, 3𝑗̂ − 𝑘̂, 2𝑖̂ + 𝑗̂ − 15𝑘̂ is 546.
31. Show that the four points whose position vectors are 6𝑖̂ − 7𝑗̂, 16𝑖̂ − 19𝑗̂ −
4𝑘̂, 3𝑗̂ − 6𝑘̂, 2𝑖̂ − 5𝑗̂ + 10𝑘̂ are coplanar.
32. Three vectors 𝑎⃗ = (12, 4, 3), 𝑏⃗⃗ = (8, −12, −9), 𝑐⃗ = (33, −4, −24) define
a parallelepiped. Evaluate the lengths of its edges, areas of its faces and its
volume.
33. Prove that (𝑏⃗⃗ + 𝑐⃗). (𝑐⃗ + 𝑎⃗) × (𝑎⃗ + 𝑏⃗⃗) = 2[𝑎⃗ 𝑏⃗⃗ 𝑐⃗].
34. If 𝑎⃗, 𝑏⃗⃗ 𝑐⃗ are the position vectors of the points A, B, C respectively, show
that 𝑏⃗⃗ × 𝑐⃗ + 𝑐⃗ × 𝑎⃗ + 𝑎⃗ × 𝑏⃗⃗ is perpendicular to plane ABC.
OR
Prove that the normal to the plane containing the three points whose position
vectors are 𝑎⃗, 𝑏⃗⃗ 𝑐⃗ lies in the direction of 𝑏⃗⃗ × 𝑐⃗ + 𝑐⃗ × 𝑎⃗ + 𝑎⃗ × 𝑏⃗⃗.
2
35. For any two vectors 𝑎⃗ 𝑎𝑛𝑑 𝑏⃗⃗, prove that (1 + |𝑎⃗|2 ) (1 + |𝑏⃗⃗| ) =
2
2
(1 − 𝑎⃗. 𝑏⃗⃗) + |𝑎⃗ + 𝑏⃗⃗ + (𝑎⃗ × 𝑏⃗⃗)| .
36. If 𝑎⃗, 𝑏⃗⃗, 𝑐⃗ 𝑎𝑟𝑒 𝑛𝑜𝑛 − 𝑐𝑜𝑝𝑙𝑎𝑛𝑎𝑟 𝑣𝑒𝑐𝑡𝑜𝑟𝑠 𝑎𝑛𝑑 𝑝⃗ = [
𝑎⃗⃗
𝑟⃗ = [
⃗⃗
𝑎⃗⃗×𝑏
𝑎⃗⃗
⃗⃗
𝑏
⃗⃗×𝑐⃗
𝑏
⃗⃗
𝑏
, 𝑞⃗ = [
]
𝑐⃗
𝑐⃗×𝑎⃗⃗
𝑎⃗⃗
⃗⃗
𝑏
,
𝑐⃗]
, prove that 𝑝⃗. (𝑎⃗ + 𝑏⃗⃗) + 𝑞⃗. (𝑏⃗⃗ + 𝑐⃗) + 𝑟⃗. (𝑐⃗ + 𝑎⃗) = 3.
𝑐⃗]
37. Let 𝑎⃗, 𝑏⃗⃗, 𝑐⃗ be three non-zero vectors. If 𝑎⃗. (𝑏⃗⃗ × 𝑐⃗) = 0 and 𝑏⃗⃗ 𝑎𝑛𝑑 𝑐⃗ are nonparallel vectors, then prove that 𝑎⃗ = 𝜆𝑏⃗⃗ + 𝜇𝑐⃗ where 𝜆 𝑎𝑛𝑑 𝜇 are some
scalars.
38. Let 𝑎⃗, 𝑏⃗⃗, 𝑐⃗
be three non-zero vectors such that 𝑐⃗ is a unit vector
𝜋
perpendicular to both 𝑎⃗ 𝑎𝑛𝑑 𝑏⃗⃗. If the angle between 𝑎⃗ 𝑎𝑛𝑑 𝑏⃗⃗ is , prove that
6
[𝑎⃗
𝑏⃗⃗
2
1
2
2
𝑐⃗] = 4 |𝑎⃗| |𝑏⃗⃗| .
39. If the vectors 𝛼⃗ = 𝑎𝑖̂ + 𝑗̂ + 𝑘̂, 𝛽⃗ = 𝑖̂ + 𝑏𝑗̂ + 𝑘̂ 𝑎𝑛𝑑 𝛾⃗ = 𝑖̂ + 𝑗̂ + 𝑐𝑘̂
coplanar, then prove that
1
1−𝑎
+
1
1−𝑏
+
1
1−𝑐
are
= 1 where 𝑎 ≠ 1, 𝑏 ≠ 1, 𝑐 ≠ 1.
THREE DIMENSSIONAL GEOMETRY
Section-A(each question carries 1 mark)
1. Show that the points A(2, 3, -4), B(1, -2, 3) and C(3, 8, -11) are collinear.
2. Find the direction cosines of a line which makes equal angles with the
coordinate axes.
3. If a line has the direction ratios -18, 12, -4, then what are its direction cosines?
4. Find the direction cosines of the line represented by x=2-2t, y=3+t, z=4-5t.
5. Find the direction cosines of the line 2x-5=3y-7=4z-8.
6. Find the vector equation for the line through the points A(3, 4, -7) and B(1, -1,
6).
7. If a line makes angles 900 , 1350 , 𝑎𝑛𝑑 450 with the positive directions of X, Y
and Z axes, find its direction cosines.
8. If a line makes angles 𝛼, 𝛽 𝑎𝑛𝑑 𝛾 with the coordinate axes, prove that
sin2 𝛼 + sin2 𝛽 + sin2 𝛾 = 2.
9. A straight line makes angles 600 𝑎𝑛𝑑 450 with the positive direction of X-axis
and Y-axis respectively. What angle does it make with the Z-axis?
10. Show that the line joining the points A(7, 8, 6) and B(9, 11, 7) is
perpendicular to the line joining the points C(9, 1, -6) and D(7, 2, -5).
11. Using direction numbers, show that the points A(-2, 4, 7), B(3, -6, -8) and C(1,
-2, -2) are collinear.
12. If the Cartesian equations of a line are
𝑥−1
2
=
𝑦+2
3
=
𝑧−5
−1
, find its vector
equation.
13. Find the direction cosines of the perpendicular from the origin to the plane
𝑟⃗. (6𝑖̂ − 3𝑗̂ − 2𝑘̂) + 1 = 0.
14. Find the vector equation of the plane whose Cartesian equation is 3x7y+9z+12=0.
Section-B(each question carries 4 marks)
1. Find 𝜆 in order that the line joining the points A(3, 2, 1) and B(4, 𝜆, 5) may
intersect the line joining C(4, 2, -2) and D(6, 5, -1).
2. For what value of p and q will the line joining points A(3, 2, 5) and B(p, 5, 0) be
parallel to the line joining points C(1, 3, q) and D(6, 4, -1).
3. Find the direction cosines of the line which is perpendicular to the lines with
direction cosines proportional to <1, -2, -2> and <0, 2, 1>.
4. Find the angle between the two lines whose direction cosines are given by the
equations l + m + n = 0, l2 + m2 + n2 = 0.
5. Find the coordinates of the foot of the perpendicular drawn from the point
A(1, 8, 4) to the line joining the points B(0, -1, 3) and C(2, -3, -1).
6. Find the point in which the join of A(-9, 4, 5) and B(11, 0, -1) is met by the
perpendicular from the origin.
7. Show that the lines
𝑥−1
2
=
𝑦−2
3
=
𝑧−3
𝑎𝑛𝑑
4
𝑥−4
5
=
𝑦−1
= 𝑧 intersect. Find also
2
the point of intersection of the lines.
8. Find the points on the line through the points A(1,2,3) and B(3,5,9) at a
distance of 14 units from the mid-point of segment AB.
9. Find the values of p and q by using vector method such that the points A(5, 0,
5), B(2, 1, 3) and C(-4, p, q) are collinear.
10. Find the coordinates of the point where the line through the points (5, 1, 6)
and (3,4,1) crosses the yz-plane.
11. Find the value of p so that the lines
1−𝑥
3
=
7𝑦−14
2𝑝
=
𝑧−3
and
2
7−7𝑥
=
3𝑝
𝑦−5
1
=
6−𝑧
5
are perpendicular to each other.
12. Find the angle between the lines
𝑥+1
5
=
𝑦−2
−2
=
𝑧−1
2
13. Find the coordinates of the point where the line
and
𝑥+1
2
𝑥+3
−2
=
=
𝑦+2
3
𝑧−4
=
3
, y=-5.
𝑧+3
4
meets the
plane x+y+4z=6.
14. Show that the line 𝑟⃗ = 2𝑖̂ + 3𝑗̂ + 𝜆(7𝑖̂ − 5𝑘̂) lies in the plane 𝑟⃗. (5𝑖̂ − 3𝑗̂ +
7𝑘̂) = 1.
15. Show that the line 𝑟⃗ = 4𝑖̂ − 7𝑘̂ + 𝜆(4𝑖̂ − 2𝑗̂ + 3𝑘̂) is parallel to the plane
𝑟⃗. (5𝑖̂ + 4𝑗̂ − 4𝑘̂) = 7.
16. Find the vector equation of the straight line passing through (1, 2, 3) and
perpendicular to the plane 𝑟⃗. (𝑖̂ + 2𝑗̂ − 5𝑘̂) + 9 = 0.
17. If the line drawn from the point (-10, 5, 4) meets a plane at right angles at
the point (4, -1, 2), find the equation of the plane.
18. The position vectors of two points A and B are 3𝑖̂ + 𝑗̂ + 2𝑘̂ and 𝑖̂ − 2𝑗̂ − 4𝑘̂
respectively. Find the vector equation of the plane passing through B and
perpendicular to the vector ⃗⃗⃗⃗⃗⃗
𝐴𝐵.
19. Find the vector and the Cartesian equations of the plane passing through the
point (5, 2, -4) and perpendicular to the line with direction ratios <2, 3, -1>.
20. Find the equation of the plane such that the length of the perpendicular
from the origin to the plane is 5 units and this perpendicular makes angles of
600 𝑎𝑛𝑑 450 with X-axis and Y-axis respectively.
Section-C(each question carries 6 marks)
1. Find the shortest distance between the two lines whose vector equations are
given by
𝑟⃗ = 𝑖̂ + 2𝑗̂ + 𝑘̂ + 𝜆(𝑖̂ − 𝑗̂ + 𝑘̂) 𝑎𝑛𝑑 𝑟⃗ = 2𝑖̂ − 𝑗̂ − 𝑘̂ + 𝜇(2𝑖̂ + 𝑗̂ + 2𝑘̂).
2. Find the shortest distance between the two lines
𝑦
−2
=
𝑧+1
−2
𝑥−6
1
𝑦−2
=
=
−2
𝑧−2
2
and
𝑥+4
3
=
.
Also find the equations of the line of shortest distance and the co-ordinates of
the points where it meets the given lines.
3. Show that the lines
𝑥+1
3
=
𝑦+3
3
=
𝑧+5
7
and
𝑥−2
1
=
𝑦−4
3
=
𝑧−6
intersect. Find
5
their point of intersection. Also find the equation of the plane containing these
lines.
4. Find the direction cosines of the four diagonals and the angle between two
diagonals of a cube.
5. If the two straight lines whose direction cosines are given by al + bm + cn = 0
𝑓
𝑔
ℎ
𝑎
𝑏
𝑐
and fmn + gnl + hlm = 0 are perpendicular, then show that + + = 0.
6. If a line L makes angles 𝛼, 𝛽, 𝛾, 𝛿 with the four diagonals of a cube, prove that
4
cos 2 𝛼 + cos 2 𝛽 + cos 2 𝛾 + cos 2 𝛿 = .
3
7. Find the foot of the perpendicular from P(1, 2, 3) on the line
𝑥−6
3
=
𝑦−7
2
=
𝑧−7
−2
.
Also obtain the equations and the length of perpendicular. Also Find the image
point of P.
8. Determine the equations of the line passing through the point (-1, 3, -2) and
𝑥
𝑦
1
2
perpendicular to the lines =
=
𝑧
3
𝑎𝑛𝑑
𝑥+2
−3
=
𝑦−1
2
=
𝑧+1
5
.
9. Find the equations of the two lines through the origin which intersect the line
𝑥−3
2
=
𝑦−3
1
=
𝑧
1
𝜋
at angles of .
3
10. A line passes through the point (-1, 3, 7) and is perpendicular to the lines 𝑟⃗ =
(2𝑖̂ − 3𝑗̂) + 𝜆(2𝑖̂ − 3𝑗̂ + 𝑘̂) and 𝑟⃗ = (𝑖̂ − 𝑗̂ + 𝑘̂) + 𝜇(7𝑗̂ − 5𝑘̂) . Obtain its
equation.
11. Find the vector equation of the line passing through the point (2, 3, 2) and
parallel to the line 𝑟⃗ = −2𝑖̂ + 3𝑗̂ + 𝜆(2𝑖̂ − 3𝑗̂ + 6𝑘̂). Also find the distance
between these lines.
12. Find the equation of the line which intersects the lines
𝑥+2
1
=
𝑦−3
2
=
𝑧+1
4
𝑥−1
2
=
𝑦−2
3
=
𝑧−3
4
and
and passes through the point (1, 1, 1).
13. Find the image of the point (1, 2, 3) in the plane x+2y+4z=38.
14. Find the length and the foot of the perpendicular from the point (1, 1, 2) to
the plane 2x-2y+4z+5=0. Also find the image point of (1, 1, 2).
15. Find the distance of the point (1, -2, 3) from the plane x – y + z = 5 measured
𝑥
𝑦
2
3
along a line parallel to =
=
𝑧
−6
.
PREPARED BY DR BANAMALI JENA, VP AND DR RAJENDRA KUMAR DHAL
JNV KHIRIYADEVAT, ASHOKNAGAR, MP AND JNV PARALAKHEMUNDI, ODISHA