VECTORS:
1) Find the values of 𝑥 and 𝑦 so that 2𝑖⃗ + 3𝑗⃗ and 𝑥𝑖⃗ + 𝑦𝑗⃗ are equal.
2) For given vectors, 𝑎⃗ = 2𝑖̂ − 𝑗̂ + 2𝑘̂ and 𝑏⃗⃗ = −𝑖̂ + 𝑗̂ − 𝑘̂ , find the unit vectors in the
direction of the vector 𝑎⃗ + 𝑏⃗⃗.
3) Show that the vector = 𝑖̂ + 𝑗̂ + 𝑘̂ is equally inclined to the axes OX, OY and OZ
4) Show that the points A, B and C with position vectors 𝑎⃗ = 3𝑖̂ − 4𝑗̂ − 4𝑘̂, 𝑏⃗⃗ = 2𝑖̂ −
𝑗̂ + 𝑘̂ and 𝑐⃗ = 𝑖̂ − 3𝑗̂ − 5𝑘̂, respectively form the vertices of a right angled triangle.
5) If 𝑎⃗ = 5𝑖̂ − 𝑗̂ − 3𝑘̂ and 𝑏⃗⃗ = 𝑖̂ + 3𝑗̂ − 5𝑘̂ , then show that the vectors 𝑎⃗ + 𝑏⃗⃗ and 𝑎⃗ − 𝑏⃗⃗
are perpendicular.
6) Find the projection of the vector 𝑎⃗ = 2𝑖̂ + 3𝑗̂ + 2𝑘̂ on the vector 𝑏⃗⃗ = 𝑖̂ + 2𝑗̂ + 𝑘̂
7) If 𝑎⃗ is a unit vector and (𝑥⃗ − 𝑎⃗). (𝑥⃗ + 𝑎⃗)= 8, then find |𝑥⃗ |
8) Find the angle between two vectors 𝑎⃗ and 𝑏⃗⃗ with magnitude √3 and 2, respectively
having 𝑎⃗. 𝑏⃗⃗ = √6
9) If 𝑎⃗ = 2𝑖̂ + 2𝑗̂ + 3𝑘̂ , 𝑏⃗⃗ = −𝑖̂ + 2𝑗̂ + 𝑘̂ and 𝑐⃗ = 3𝑖̂ + 𝑗̂ are such that 𝑎⃗ + µ𝑏⃗⃗ is
perpendicular to 𝑐⃗, then find the value of µ.
10) If 𝑎⃗, 𝑏⃗⃗, 𝑐⃗ are unit vectors such that 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ = ⃗0⃗, find the value of 𝑎⃗. 𝑏⃗⃗ + 𝑏⃗⃗. 𝑐⃗ + 𝑐⃗. 𝑎⃗
̂ − 5𝑘̂ and 3𝑖̂ − 4𝑗̂ − 4𝑘̂ form the vertices of
11) Show that the vector 2𝑖̂ − 𝑗̂ + 𝑘̂ , 𝑖̂ − 3𝑗
a right angled triangle.
12) Find a unit vector perpendicular to each of the vector 𝑎⃗ + 𝑏⃗⃗ and 𝑎⃗ − 𝑏⃗⃗, where 𝑎⃗ =
3𝑖̂ + 2𝑗̂ + 2𝑘̂ and 𝑏⃗⃗ = 𝑖̂ + 2𝑗̂ − 2𝑘̂ .
𝜋
𝜋
13) If a unit vector 𝑎⃗ makes an angles with 𝑖̂ , with 𝑗̂ and an acute angle 𝜃 with 𝑘̂ ,
3
4
then find 𝜃 and hence, the components of 𝑎⃗.
14) Find the area of the parallelogram whose adjacent sides are determined by the
̂ + 𝑘̂
vectors 𝑎⃗ = 𝑖̂ − 𝑗̂ + 3𝑘̂ and 𝑏⃗⃗ = 2𝑖̂ − 7𝑗
⃗⃗. Evaluate the quantity
15) Three vectors 𝑎⃗, 𝑏⃗⃗ and 𝑐⃗ satisfy the condition 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ = 0
𝜇 = 𝑎⃗. 𝑏⃗⃗ + 𝑏⃗⃗. 𝑐⃗ + 𝑐⃗. 𝑎⃗ , if |𝑎⃗| = 1, |𝑏⃗⃗| = 4 and |𝑐⃗| = 2
16) If with reference to the right handed system of mutually perpendicular unit vectors
⃗⃗⃗⃗⃗1 + 𝛽
⃗⃗⃗⃗⃗2 ,
𝑖̂, 𝑗̂ and 𝑘̂ , 𝛼⃗ = 3𝑖̂ − 𝑗̂, 𝛽⃗ = 2𝑖̂ + 𝑗̂ − 3𝑘̂ , then express 𝛽⃗ in the form 𝛽⃗ = 𝛽
where ⃗⃗⃗⃗⃗
𝛽1 is parallel to 𝛼⃗ and ⃗⃗⃗⃗⃗
𝛽2 is perpendicular to 𝛼⃗.
Prepared by Mr. Anirban Nayak
PGT, Mathematics, DPS Jodhpur
Mobile No.- 9828353006
17) Find a vector of magnitude 5 units, and parallel to the resultant of the vectors 𝑎⃗ =
2𝑖̂ + 3𝑗̂ − 𝑘̂ and 𝑏⃗⃗ = 𝑖̂ − 2𝑗̂ + 𝑘̂
18) 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
19) Show that the direction cosines of a vector equally inclined to the axes OX, OY and
OZ are
1
,
1
,
1
√3 √3 √3
.
20) Let 𝑎⃗ = 𝑖̂ + 4𝑗̂ + 2𝑘̂ , 𝑏⃗⃗ = 3𝑖̂ − 2𝑗̂ + 4𝑘̂ and 𝑐⃗ = 2𝑖̂ − 𝑗̂ + 4𝑘̂ . Find a vector 𝑑⃗ which is
perpendicular to both 𝑎⃗ and 𝑏⃗⃗, and 𝑐⃗. 𝑑⃗ = 15
21) The scalar product of the vector 𝑖̂ + 𝑗̂ + 𝑘̂ with a unit vector along the sum of
vectors 2𝑖̂ + 4𝑗̂ − 5𝑘̂ and 𝜇𝑖̂ + 2𝑗̂ + 3𝑘̂ is equal to one. Find the value of 𝜇.
22) If 𝑎⃗, 𝑏⃗⃗, 𝑐⃗ are mutually perpendicular vectors of equal magnitudes, show that the
vector 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ is equally inclined to 𝑎⃗, 𝑏⃗⃗ and 𝑐⃗
23) Let 𝑎⃗ and 𝑏⃗⃗ be two unit vectors and 𝜃 is the angle between them. If 𝑎⃗ + 𝑏⃗⃗ is a unit
vector , then find the value of 𝜃
24) A girl walks 4 km towards west, then she walks 3 km in a direction 300 east of north
and stops. Determine the girl’s displacement from her initial point of departure.
25) Let 𝑎⃗, 𝑏⃗⃗ and 𝑐⃗ be three vectors such that |𝑎⃗| = 3, |𝑏⃗⃗| = 4, |𝑐⃗| = 5 and each one of
them being perpendicular to the sum of the other two, find |𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗|.
EXTRA Problems (Part-1)
1) Three vectors 𝑎⃗, 𝑏⃗⃗ and 𝑐⃗ satisfy the condition 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ = ⃗0⃗. Evaluate the quantity
𝜇 = 𝑎⃗. 𝑏⃗⃗ + 𝑏⃗⃗. 𝑐⃗ + 𝑐⃗. 𝑎⃗, if |𝑎⃗|= 1, |𝑏⃗⃗|= 2 and |𝑐⃗|= 2
2
2
2) For any two vectors 𝑎⃗ and 𝑏⃗⃗, show that (1 + |𝑎⃗|2 ) (1 + |𝑏⃗⃗| )= (1 − 𝑎⃗. 𝑏⃗⃗) +|𝑎⃗ +
2
⃗⃗⃗⃗
𝑏 + (𝑎⃗ × 𝑏⃗⃗)|
3) If the vectors 𝑎𝑖̂ + 𝑎𝑗̂ + 𝑐𝑘̂ , 𝑖̂ + 𝑘̂ and 𝑐𝑖̂ + 𝑐𝑗̂ + 𝑏𝑘̂ are coplanar, show that 𝑐 2 = 𝑎𝑏
4) If 𝑎⃗ = 𝑖̂ + 𝑗̂ + 𝑘̂ and 𝑏⃗⃗ = 𝑗̂ − 𝑘̂ , find a vector 𝑐⃗ such that 𝑎⃗ × 𝑐⃗ = 𝑏⃗⃗ and 𝑎⃗. 𝑐⃗ = 3
5) If 𝑎⃗ × 𝑏⃗⃗ = 𝑐⃗ × 𝑑⃗ and 𝑎⃗ × 𝑐⃗ = 𝑏⃗⃗ × 𝑑⃗, show that (𝑎⃗ − 𝑑⃗)is parallel to (𝑏⃗⃗ − 𝑐⃗), it is
being given that 𝑎⃗ ≠ 𝑑⃗ and 𝑏⃗⃗ ≠ 𝑐⃗
Prepared by Mr. Anirban Nayak
PGT, Mathematics, DPS Jodhpur
Mobile No.- 9828353006
6) If the points A(2,𝛽,3), B(𝛼, −5, 1) and C( -1, 11, 9) are collinear, find the values of 𝛼
and 𝛽 by vector method.
7) If 𝑎⃗ = 2𝑖̂ − 𝑗̂ + 𝑘̂ , 𝑏⃗⃗ = 𝑖̂ + 3𝑗̂ − 𝑘̂ , 𝑐⃗ = 2𝑖̂ + 𝑗̂ − 3𝑘̂ and 𝑑⃗ = 3𝑖̂ + 2𝑗̂ + 5𝑘̂, find
scalars 𝛼, 𝛽 and 𝛾 such that 𝑑⃗ = 𝛼𝑎⃗ + 𝛽𝑏⃗⃗ + 𝛾𝑐⃗
8) If the vectors 𝑎𝑖̂ + 𝑗̂ + 𝑘̂ , 𝑖̂ + 𝑏𝑗̂ + 𝑘̂ and 𝑖̂ + 𝑗̂ + 𝑐𝑘̂ are coplanar, find the value of
1
1−𝑎
+
1
1−𝑏
+
1
1−𝑐
9) If 𝑎⃗, 𝑏⃗⃗, 𝑐⃗ are the unit vectors such that 𝑎⃗. 𝑏⃗⃗ = 𝑎⃗. 𝑐⃗ = 0 and the angle between 𝑏⃗⃗ and
𝜋
𝑐⃗ is , then prove that 𝑎⃗ = ±(𝑏⃗⃗ × 𝑐⃗)
6
10) If 𝑎⃗. 𝑏⃗⃗ = 𝑎⃗. 𝑐⃗, 𝑎⃗ × 𝑏⃗⃗ = 𝑎⃗ × 𝑐⃗ and 𝑎⃗ ≠ ⃗0⃗, then prove that 𝑏⃗⃗ = 𝑐⃗
11) If 𝑎⃗ = 𝑖̂ + 𝑗̂ + 𝑘̂ , 𝑐⃗ = 𝑗̂ − 𝑘̂ are given vectors, then find a vector 𝑏⃗⃗ satisfying the
equations 𝑎⃗ × 𝑏⃗⃗ = 𝑐⃗ and 𝑎⃗. 𝑏⃗⃗ = 3
12) If three vectors 𝑎⃗, 𝑏⃗⃗ and 𝑐⃗ satisfy the condition 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ = ⃗0⃗, then prove that 𝑎⃗ ×
𝑏⃗⃗ = 𝑏⃗⃗ × 𝑐⃗ = 𝑐⃗ × 𝑎⃗
EXTRA Problems(Part- Ii)
1) The scalar product of the vector 𝑎⃗ = 𝑖̂ + 𝑗̂ + 𝑘̂ with a unit vector along the sum of
vectors 𝑏⃗⃗ = 2𝑖̂ + 4𝑗̂ − 5𝑘̂ and 𝑐⃗ = 𝑖̂ + 2𝑗̂ + 3𝑘̂ is equal to one. Find the value of 
and hence find the unit vector along 𝑏⃗⃗ + 𝑐⃗
2) Prove that [𝑎⃗ + 𝑏⃗⃗ 𝑏⃗⃗ + 𝑐⃗ 𝑐⃗ + 𝑎⃗] = 𝑘[𝑎⃗ 𝑏⃗⃗ 𝑐⃗]
3) Suppose 𝑎⃗ = 𝑖̂ − 7𝑗̂ + 3𝑘̂ , 𝑏⃗⃗ = 𝑖̂ + 𝑗̂ + 2𝑘̂ . If the angle between 𝑎⃗ and 𝑏⃗⃗ is
greater than 900, then prove that  satisfies the inequality – 7< <1.
4) Let 𝑣⃗ = 𝑖̂ + 𝑗̂ − 𝑘̂ and 𝑤
⃗⃗⃗ = 𝑖̂ + 3𝑘̂ . If 𝑢̂ is a unit vector , then find the maximum
value of the scalar triple product 𝑢̂, 𝑣⃗, 𝑤
⃗⃗⃗
2
5) For any vector 𝑎⃗ , prove that (𝑎⃗ × 𝑖̂)2 + (𝑎⃗ × 𝑗̂)2 + (𝑎⃗ × 𝑘̂ ) = 2𝑎⃗2
6) If 𝑎⃗ = 𝑖̂ − 𝑘̂ , 𝑏⃗⃗ = 𝑥𝑖̂ + 𝑗̂ + (1 − 𝑥)𝑘̂ and 𝑐⃗ = 𝑦𝑖̂ + 𝑥𝑗̂ + (1 + 𝑥 − 𝑦)𝑘̂ , then prove
that [𝑎⃗ 𝑏⃗⃗ 𝑐⃗] depends upon neither 𝑥 nor 𝑦.
7) If 𝑎⃗, 𝑏⃗⃗, 𝑐⃗ are mutually perpendicular vectors of equal magnitudes 𝑥, show that the
vector 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ is equally inclined to 𝑎⃗, 𝑏⃗⃗ and 𝑐⃗. Also find the angle.
Prepared by Mr. Anirban Nayak
PGT, Mathematics, DPS Jodhpur
Mobile No.- 9828353006
⃗⃗ , |𝑎⃗| = |𝑏⃗⃗| = |𝑐⃗| and 𝜃 is the angle between 𝑏⃗⃗ and 𝑐⃗, then find
8) If 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ = 0
the value of 𝑐𝑜𝑠𝑒𝑐 2 𝜃 + 𝑐𝑜𝑡 2 𝜃, where 0≤ 𝜃 ≤ 𝜋.
⃗⃗ . If |𝛼⃗| = 3,
9) If 𝛼⃗ , 𝛽⃗ and 𝛾⃗ are three vectors satisfying the condition 𝛼⃗ + 𝛽⃗ + 𝛾⃗ = 0
|𝛽⃗ | = 4 and |𝛾⃗ | = 5, show that 𝛼⃗. 𝛽⃗ + 𝛽⃗. 𝛾⃗ + 𝛾⃗. 𝛼⃗ = −25.
10) If 𝛼⃗ , 𝛽⃗ and 𝛾⃗ are three vectors satisfying the condition 𝛼⃗ + 𝛽⃗ + 𝛾⃗ = ⃗0⃗ . If |𝛼⃗| = 3,
|𝛽⃗ | = 5 and |𝛾⃗ | = 7; find the angle between 𝛼⃗ and 𝛽⃗ .
11) Prove that [𝑖̂ 𝑗̂ 𝑘̂ ] + [𝑗̂ 𝑘̂ 𝑖̂] + [𝑘̂ 𝑖̂ 𝑗] = 3
12) If four points A(𝑎⃗), B(𝑏⃗⃗), C(𝑐⃗) and D(𝑑⃗) are coplanar, then prove that [𝑎⃗ 𝑏⃗⃗ 𝑐⃗]=
[𝑏⃗⃗ 𝑐⃗ 𝑑⃗] + [𝑐⃗ 𝑎⃗ 𝑑⃗] + [𝑎⃗ 𝑏⃗⃗ 𝑑⃗]
𝜃
13) If 𝑎̂ and 𝑏̂ are unit vectors inclined at an angle 𝜃, then prove that tan =
2
|𝑎̂− 𝑏̂|
.
|𝑎̂+ 𝑏̂|
14) If 𝑎⃗, 𝑏⃗⃗, 𝑐⃗ are mutually perpendicular unit vectors, then prove that |𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗| = √3
15) If |𝑎⃗ + 𝑏⃗⃗| = 60, |𝑎⃗ − 𝑏⃗⃗| = 40 and |𝑏⃗⃗| = 46, Find |𝑎⃗|
16) If the sum of two unit vectors is a unit vector. Prove that the magnitude of their
difference is √3
𝜋
𝜋
4
2
17) Find a vector 𝑎⃗ of magnitude 3√2 units which makes an angle of and with 𝑦 and
𝑧 − axes, respectively.
Prepared by Mr. Anirban Nayak
PGT, Mathematics, DPS Jodhpur
Mobile No.- 9828353006