Maths Assignment On Vectors
Class : XII
Amrita vidyalayam chennai
Due date : 04/6/2015
⃗⃗⃗ = 𝟐𝒊 −
1. Find the unit vector in the direction of the sum of the vectors 𝒂
⃗ 𝒂𝒏𝒅 ⃗𝒃 = −𝒊 + 𝒋 + 𝟑𝒌
⃗
𝒋 + 𝟐𝒌
⃗⃗⃗⃗⃗⃗ ,
2. Find a vector of magnitude 11 in the direction opposite to that of 𝑷𝑸
where P and Q are the points (1,3,2) and (-1,0,8) respectively.
⃗ of magnitude 3√𝟐 units which makes an angle of
3. Find a vector 𝒓
𝝅
𝝅
𝒂𝒏𝒅 with y and Z axes respectively.
𝟒
𝟐
4. Find the position vector of a point R which divides the line joining the
two points P and Q with position vector
⃗ , respectively in the ratio 1:2 (i)
⃗⃗⃗⃗⃗⃗
⃗ − 𝟐𝒃
𝑶𝒑 = ⃗⃗⃗⃗⃗
𝟐𝒂 + ⃗𝒃 𝒂𝒏𝒅 ⃗⃗⃗⃗⃗⃗
𝑶𝑸 = 𝒂
internally (ii) externally.
5. Find all vectors of magnitude 10√𝟑 that are perpendicular to the plane
⃗ 𝒂𝒏𝒅 −𝒊
⃗
⃗⃗⃗⃗ + 𝟑𝒋 + 𝟒𝒌
of 𝒊 + 𝟐𝒋 + 𝒌
⃗ respectively the two sides AB and AC
6. The 2 vectors 𝒊 + 𝒋 𝒂𝒏𝒅 𝒊 − 𝒋 + 𝟒𝒌
respectively of a triangle ABC. Find the length of the median through
A.
⃗ are units vectors , then what is the angle between 𝒂
⃗
⃗ 𝒂𝒏𝒅 𝒃
⃗ 𝒂𝒏𝒅 𝒃
7. If 𝒂
for √⃗⃗⃗⃗⃗
𝟑𝒂 − ⃗𝒃 to be a unit vector ?
⃗⃗⃗⃗⃗ and ⃗𝒃 = −𝒊
⃗⃗⃗⃗⃗ are sides of a
⃗⃗⃗⃗ − 𝟐𝒋
⃗⃗⃗⃗ + 𝟐𝒌
⃗ = 𝟑𝒊
⃗⃗⃗⃗ − 𝟐𝒌
8. The vectors 𝒂
parallelogram. Find the acute angle between its diagonal .
⃗ + ⃗𝒃|=|𝒂
⃗ − ⃗𝒃|, then show that 𝒂
⃗ 𝒂𝒏𝒅⃗⃗⃗𝒃 are orthogonal
9. If |𝒂
⃗ , Find value of
10.For any vector 𝒂
⃗ )𝟐
⃗ 𝑿𝒊)𝟐+(𝒂
⃗ 𝑿𝒋)𝟐 + (𝒂
⃗ 𝑿𝒌
(𝒂
11.The dot product of a vector with the vectors 2i  3 j  k , 4i  j and
i  3 j  7k are respectively. 9, 7 and 6. Find the vector.
12.
In a triangle ABC, the sides AB and BC are represented by the vectors
2i  j  2k , i  3 j  5k respectively. Find the vector representing CA.
13.
If 𝑎 and 𝑏⃗ are two vectors of magnitude 3 and
2
3
respectively such that 𝑎 x 𝑏⃗ is a unit vector, write angle
between 𝑎 and 𝑏⃗
If 𝑎 = 7𝑖̂ + 𝑗̂ -4𝑘̂ and 𝑏⃗ = 2𝑙̂ + 6𝑗̂ + 3𝑘̂ , find the projection
of 𝑎 on 𝑏⃗ .
15.
Using vectors, find the area of the triangle with vertices
A(1,1,2) , B(2,3,5) and C(1,5,5).
16.
If 𝑎 , 𝑏⃗ , 𝑐 are unit vector such that 𝑎
⃗⃗⃗ + 𝑏⃗ + 𝑐 = 𝑜 ,
find the value of
𝑎 . 𝑏⃗ + ⃗⃗⃗
𝑏 .𝑐+𝑐.𝑎.
17.
Three vectors 𝑎 , 𝑏⃗, 𝑐 , satisfy the condition 𝑎 𝑏⃗+𝑐 = 0.
Evaluate the quantity 𝜇 = 𝑎 . 𝑏⃗ + 𝑏⃗ . 𝑐⃗⃗ + 𝑐 . 𝑎 if |𝑎 | = 1, |𝑏⃗| =4
and |𝑐| = 2.
14.
 


18.If | a | 1, | b | 1, a b  cos find | a  b |
19.Write the position vector of a point dividing the line segment joining points
A and B with position vectors a and b externally in the ratio 1:4 where


a  2i  3 j  4k and b  i  j  k
20.Find the unit vector perpendicular to the plane ABC where the position
vectors A, B and C are 2i  j  k , i  j  2k and 2i  3k respectively
21.Find the projection of the vector a  i  2 j  k on the vector
b  4i  4 j  7k
22.The scalar product of the vector i  j  k with a unit vector along the sum
of vectors 2i  4 j  5k and  i  2 j  3k is equal to one. Find the value of .
 




Let a, b and c be three vectors such that a  3 , b  4 , c  5 and
23.
each one of there is perpendicular to the sum of the other two, find



a b c
  
24.If a, b, c are mutually perpendicular vectors of equal magnitudes show that


 


the vector a  b  c is equally inclined to a, b and c .
25.Find a vector in the direction of vector
units
26.
27.
Find the value of
Find
the
unit
vector
that has magnitude 7
in
the
direction
of
28.
Write the position vector of a point dividing the line segment
joining points A and B with position Vectors
externally in the
ratio 1 : 3 , where
and
.
29.
Find the direction cosines of a vector that makes equal angle
with the coordinate axes
30.Find the angle made by the vector i – 4j + 8k with the z – axis
31.
A Vector makes angles 60 and 45 with x and y-axis
respectively. Find the angle which it makes with z-axis.
32.
Find the value of
33.
Find
34.
Find the position vector of a point R which divides the line
joining the points P(i + 2j – k) and Q(-i + j + k) in the ratio 2 : 1
externally.
35.
Find the area of the parallelogram whose adjacent sides are
determined by the vectors
36.
If
37.
The scalar product of the vector i + j + k with a unit vector along
the sum of vectors 2i + 2j – 5k and
λ i + 2j + 3k is equal to
one. Find the value of λ .
38.
Find the vector of magnitude 5 units which is perpendicular to
both the vectors
39.
If the sum of two unit vectors is a unit vector, Prove that the
magnitude of their difference is
.
40.
If with reference to a right handed system of mutually
perpendicular
unit
vectors
,
we
have
and
Express
in
the
form
where
,is
parallel to
is perpendicular to
41.
. If
are any two non zero vectors.then prove that
42.
If
43.
If
Find a vector
which
is perpendicular to
44.
Dot product of a vector with the vector i + j - 3 k , i + 3 k - 2 k
and 2 i + j + 4 k are 0, 5 and 8 respectively.find vector.
45.
Dot product of vector with vectors
are
respectively -1,6 & 5 find the vector.