Class 12
Dashboard
Mathematics
Vector Algebra
NCERT Solutions
NCERT Solutions
Vector Algebra
73 Solutions
Exercise:
All Exercises
EXERCISE 10.1
EXERCISE 10.2
EXERCISE 10.3
EXERCISE 10.4
Miscellaneous Exercise on Chapter 10
Q1
EXERCISE 10.1
Represent graphically a displacement of
40
km
,
30
∘
40 \text{ km}, 30^{\circ}
40
km
,
3
0
∘
east of north.
Q2
EXERCISE 10.1
Classify the following measures as scalars and vectors.
(i)
10 kg
(ii)
2 meters north-west
(iii)
40
∘
40^{\circ}
4
0
∘
(iv)
40 watt
(v)
10
−
19
10^{-19}
1
0
−
19
coulomb
(vi)
20
m/s
2
20 \text{ m/s}^{2}
20
m/s
2
Q3
EXERCISE 10.1
Classify the following as scalar and vector quantities.
(i)
time period
(ii)
distance
(iii)
force
(iv)
velocity
(v)
work done
Q4
EXERCISE 10.1
In a square, identify the following vectors.
(i)
Coinitial
(ii)
Equal
(iii)
Collinear but not equal
Q5
EXERCISE 10.1
Answer the following as true or false.
(i)
a
⃗
\vec{a}
a
and
−
a
⃗
-\vec{a}
−
a
are collinear.
(ii)
Two collinear vectors are always equal in magnitude.
(iii)
Two vectors having same magnitude are collinear.
(iv)
Two collinear vectors having the same magnitude are equal.
Q1
EXERCISE 10.2
Compute the magnitude of the following vectors:
a
⃗
=
i
^
+
j
^
+
k
;
b
⃗
=
2
i
^
−
7
j
^
−
3
k
^
;
c
⃗
=
1
3
i
^
+
1
3
j
^
−
1
3
k
^
\vec{a}=\hat{i}+\hat{j}+k ; \quad \vec{b}=2 \hat{i}-7 \hat{j}-3 \hat{k} ; \quad \vec{c}=\frac{1}{\sqrt{3}} \hat{i}+\frac{1}{\sqrt{3}} \hat{j}-\frac{1}{\sqrt{3}} \hat{k}
a
=
i
^
+
j
^
+
k
;
b
=
2
i
^
−
7
j
^
−
3
k
^
;
c
=
3
1
i
^
+
3
1
j
^
−
3
1
k
^
Q2
EXERCISE 10.2
Write two different vectors having same magnitude.
Q3
EXERCISE 10.2
Write two different vectors having same direction.
Q4
EXERCISE 10.2
Find the values of
x
x
x
and
y
y
y
so that the vectors
2
i
^
+
3
j
^
2 \hat{i}+3 \hat{j}
2
i
^
+
3
j
^
and
x
i
^
+
y
j
^
x \hat{i}+y \hat{j}
x
i
^
+
y
j
^
are equal.
Q5
EXERCISE 10.2
Find the scalar and vector components of the vector with initial point
(
2
,
1
)
(2,1)
(
2
,
1
)
and terminal point (
−
5
,
7
-5,7
−
5
,
7
).
Q6
EXERCISE 10.2
Find the sum of the vectors
a
⃗
=
i
^
−
2
j
^
+
k
^
,
b
⃗
=
−
2
i
^
+
4
j
^
+
5
k
^
\vec{a}=\hat{i}-2 \hat{j}+\hat{k}, \vec{b}=-2 \hat{i}+4 \hat{j}+5 \hat{k}
a
=
i
^
−
2
j
^
+
k
^
,
b
=
−
2
i
^
+
4
j
^
+
5
k
^
and
c
⃗
=
i
^
−
6
j
^
−
7
k
^
\vec{c}=\hat{i}-6 \hat{j}-7 \hat{k}
c
=
i
^
−
6
j
^
−
7
k
^
.
Q7
EXERCISE 10.2
Find the unit vector in the direction of the vector
a
⃗
=
i
^
+
j
^
+
2
k
^
\vec{a}=\hat{i}+\hat{j}+2 \hat{k}
a
=
i
^
+
j
^
+
2
k
^
.
Q8
EXERCISE 10.2
Find the unit vector in the direction of vector
P
Q
→
\overrightarrow{\mathrm{PQ}}
PQ
, where P and Q are the points
(
1
,
2
,
3
)
(1,2,3)
(
1
,
2
,
3
)
and (4, 5, 6), respectively.
Q9
EXERCISE 10.2
For given vectors,
a
⃗
=
2
i
^
−
j
^
+
2
k
^
\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}
a
=
2
i
^
−
j
^
+
2
k
^
and
b
⃗
=
−
i
^
+
j
^
−
k
^
\vec{b}=-\hat{i}+\hat{j}-\hat{k}
b
=
−
i
^
+
j
^
−
k
^
, find the unit vector in the direction of the vector
a
⃗
+
b
⃗
\vec{a}+\vec{b}
a
+
b
.
Q10
EXERCISE 10.2
Find a vector in the direction of vector
5
i
^
−
j
^
+
2
k
^
5 \hat{i}-\hat{j}+2 \hat{k}
5
i
^
−
j
^
+
2
k
^
which has magnitude 8 units.
Q11
EXERCISE 10.2
Show that the vectors
2
i
^
−
3
j
^
+
4
k
^
2 \hat{i}-3 \hat{j}+4 \hat{k}
2
i
^
−
3
j
^
+
4
k
^
and
−
4
i
^
+
6
j
^
−
8
k
^
-4 \hat{i}+6 \hat{j}-8 \hat{k}
−
4
i
^
+
6
j
^
−
8
k
^
are collinear.
Q12
EXERCISE 10.2
Find the direction cosines of the vector
i
^
+
2
j
^
+
3
k
^
\hat{i}+2 \hat{j}+3 \hat{k}
i
^
+
2
j
^
+
3
k
^
.
Q13
EXERCISE 10.2
Find the direction cosines of the vector joining the points
A
(
1
,
2
,
−
3
)
\mathrm{A}(1,2,-3)
A
(
1
,
2
,
−
3
)
and
B
(
−
1
,
−
2
,
1
)
\mathrm{B}(-1,-2,1)
B
(
−
1
,
−
2
,
1
)
, directed from A to B .
Q14
EXERCISE 10.2
Show that the vector
i
^
+
j
^
+
k
^
\hat{i}+\hat{j}+\hat{k}
i
^
+
j
^
+
k
^
is equally inclined to the axes
O
X
,
O
Y
\mathrm{OX}, \mathrm{OY}
OX
,
OY
and OZ .
Q15
EXERCISE 10.2
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are
i
^
+
2
j
^
−
k
^
\hat{i}+2 \hat{j}-\hat{k}
i
^
+
2
j
^
−
k
^
and
−
i
^
+
j
^
+
k
^
-\hat{i}+\hat{j}+\hat{k}
−
i
^
+
j
^
+
k
^
respectively, in the ratio 2 : 1
(i)
internally
(ii)
externally
Q16
EXERCISE 10.2
Find the position vector of the mid point of the vector joining the points
P
(
2
,
3
,
4
)
\mathrm{P}(2,3,4)
P
(
2
,
3
,
4
)
and
Q
(
4
,
1
,
−
2
)
\mathrm{Q}(4,1,-2)
Q
(
4
,
1
,
−
2
)
.
Q17
EXERCISE 10.2
Show that the points
A
,
B
\mathrm{A}, \mathrm{B}
A
,
B
and C with position vectors,
a
⃗
=
3
i
^
−
4
j
^
−
4
k
^
\vec{a}=3 \hat{i}-4 \hat{j}-4 \hat{k}
a
=
3
i
^
−
4
j
^
−
4
k
^
,
b
⃗
=
2
i
^
−
j
^
+
k
^
\vec{b}=2 \hat{i}-\hat{j}+\hat{k}
b
=
2
i
^
−
j
^
+
k
^
and
c
⃗
=
i
^
−
3
j
^
−
5
k
^
\vec{c}=\hat{i}-3 \hat{j}-5 \hat{k}
c
=
i
^
−
3
j
^
−
5
k
^
, respectively form the vertices of a right angled triangle.
Q18
EXERCISE 10.2
In triangle ABC, which of the following is not true:
(A)
A
B
→
+
B
C
→
+
C
A
→
=
0
→
\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}+\overrightarrow{\mathrm{CA}}=\overrightarrow{0}
AB
+
BC
+
CA
=
0
(B)
A
B
→
+
B
C
→
−
A
C
→
=
0
→
\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}-\overrightarrow{\mathrm{AC}}=\overrightarrow{0}
AB
+
BC
−
AC
=
0
(C)
A
B
→
+
B
C
→
−
A
C
→
=
0
→
\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}-\overrightarrow{\mathrm{AC}}=\overrightarrow{0}
AB
+
BC
−
AC
=
0
(D)
A
B
→
−
C
B
→
+
C
A
→
=
0
→
\overrightarrow{\mathrm{AB}}-\overrightarrow{\mathrm{CB}}+\overrightarrow{\mathrm{CA}}=\overrightarrow{0}
AB
−
CB
+
CA
=
0
Q19
EXERCISE 10.2
If
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
are two collinear vectors, then which of the following are incorrect:
(A)
b
⃗
=
λ
a
⃗
\vec{b}=\lambda \vec{a}
b
=
λ
a
, for some scalar
λ
\lambda
λ
(B)
a
⃗
=
±
b
⃗
\vec{a}= \pm \vec{b}
a
=
±
b
(C) the respective components of
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
are not proportional
(D) both the vectors
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
have same direction, but different magnitudes.
Q1
EXERCISE 10.3
Find the angle between two vectors
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
with magnitudes
3
\sqrt{3}
3
and 2 , respectively having
a
⃗
⋅
b
⃗
=
6
\vec{a} \cdot \vec{b}=\sqrt{6}
a
⋅
b
=
6
.
Q2
EXERCISE 10.3
Find the angle between the vectors
i
^
−
2
j
^
+
3
k
^
\hat{i}-2 \hat{j}+3 \hat{k}
i
^
−
2
j
^
+
3
k
^
and
3
i
^
−
2
j
^
+
k
^
3 \hat{i}-2 \hat{j}+\hat{k}
3
i
^
−
2
j
^
+
k
^
Q3
EXERCISE 10.3
Find the projection of the vector
i
^
−
j
^
\hat{i}-\hat{j}
i
^
−
j
^
on the vector
i
^
+
j
^
\hat{i}+\hat{j}
i
^
+
j
^
.
Q4
EXERCISE 10.3
Find the projection of the vector
i
^
+
3
j
^
+
7
k
^
\hat{i}+3 \hat{j}+7 \hat{k}
i
^
+
3
j
^
+
7
k
^
on the vector
7
i
^
−
j
^
+
8
k
^
7 \hat{i}-\hat{j}+8 \hat{k}
7
i
^
−
j
^
+
8
k
^
.
Q5
EXERCISE 10.3
Show that each of the given three vectors is a unit vector:
1
7
(
2
i
^
+
3
j
^
+
6
k
^
)
,
1
7
(
3
i
^
−
6
j
^
+
2
k
^
)
,
1
7
(
6
i
^
+
2
j
^
−
3
k
^
)
\frac{1}{7}(2 \hat{i}+3 \hat{j}+6 \hat{k}), \quad \frac{1}{7}(3 \hat{i}-6 \hat{j}+2 \hat{k}), \quad \frac{1}{7}(6 \hat{i}+2 \hat{j}-3 \hat{k})
7
1
(
2
i
^
+
3
j
^
+
6
k
^
)
,
7
1
(
3
i
^
−
6
j
^
+
2
k
^
)
,
7
1
(
6
i
^
+
2
j
^
−
3
k
^
)
Also, show that they are mutually perpendicular to each other.
Q6
EXERCISE 10.3
Find
∣
a
⃗
∣
|\vec{a}|
∣
a
∣
and
∣
b
⃗
∣
|\vec{b}|
∣
b
∣
, if
(
a
⃗
+
b
⃗
)
⋅
(
a
⃗
−
b
⃗
)
=
8
(\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})=8
(
a
+
b
)
⋅
(
a
−
b
)
=
8
and
∣
a
⃗
∣
=
8
∣
b
⃗
∣
|\vec{a}|=8|\vec{b}|
∣
a
∣
=
8∣
b
∣
.
Q7
EXERCISE 10.3
Evaluate the product
(
3
a
⃗
−
5
b
⃗
)
⋅
(
2
a
⃗
+
7
b
⃗
)
(3 \vec{a}-5 \vec{b}) \cdot(2 \vec{a}+7 \vec{b})
(
3
a
−
5
b
)
⋅
(
2
a
+
7
b
)
.
Q8
EXERCISE 10.3
Find the magnitude of two vectors
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
, having the same magnitude and such that the angle between them is
60
∘
60^{\circ}
6
0
∘
and their scalar product is
1
2
\frac{1}{2}
2
1
.
Q9
EXERCISE 10.3
Find
∣
x
⃗
∣
|\vec{x}|
∣
x
∣
, if for a unit vector
a
⃗
\vec{a}
a
,
(
x
⃗
−
a
⃗
)
⋅
(
x
⃗
+
a
⃗
)
=
12
(\vec{x}-\vec{a}) \cdot(\vec{x}+\vec{a})=12
(
x
−
a
)
⋅
(
x
+
a
)
=
12
.
Q10
EXERCISE 10.3
If
a
⃗
=
2
i
^
+
2
j
^
+
3
k
^
,
b
⃗
=
−
i
^
+
2
j
^
+
k
^
\vec{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=-\hat{i}+2 \hat{j}+\hat{k}
a
=
2
i
^
+
2
j
^
+
3
k
^
,
b
=
−
i
^
+
2
j
^
+
k
^
and
c
⃗
=
3
i
^
+
j
^
\vec{c}=3 \hat{i}+\hat{j}
c
=
3
i
^
+
j
^
are such that
a
⃗
+
λ
b
⃗
\vec{a}+\lambda \vec{b}
a
+
λ
b
is perpendicular to
c
⃗
\vec{c}
c
, then find the value of
λ
\lambda
λ
.
Q11
EXERCISE 10.3
Show that
∣
a
⃗
∣
b
⃗
+
∣
b
⃗
∣
a
⃗
|\vec{a}| \vec{b}+|\vec{b}| \vec{a}
∣
a
∣
b
+
∣
b
∣
a
is perpendicular to
∣
a
⃗
∣
b
⃗
−
∣
b
⃗
∣
a
⃗
|\vec{a}| \vec{b}-|\vec{b}| \vec{a}
∣
a
∣
b
−
∣
b
∣
a
, for any two nonzero vectors
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
.
Q12
EXERCISE 10.3
If
a
⃗
⋅
a
⃗
=
0
\vec{a} \cdot \vec{a}=0
a
⋅
a
=
0
and
a
⃗
⋅
b
⃗
=
0
\vec{a} \cdot \vec{b}=0
a
⋅
b
=
0
, then what can be concluded about the vector
b
⃗
\vec{b}
b
?
Q13
EXERCISE 10.3
If
a
⃗
,
b
⃗
,
c
⃗
\vec{a}, \vec{b}, \vec{c}
a
,
b
,
c
are unit vectors such that
a
⃗
+
b
⃗
+
c
⃗
=
0
→
\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}
a
+
b
+
c
=
0
, find the value of
a
⃗
⋅
b
⃗
+
b
⃗
⋅
c
⃗
+
c
⃗
⋅
a
⃗
\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}
a
⋅
b
+
b
⋅
c
+
c
⋅
a
.
Q14
EXERCISE 10.3
If either vector
a
⃗
=
0
→
\vec{a}=\overrightarrow{0}
a
=
0
or
b
⃗
=
0
→
\vec{b}=\overrightarrow{0}
b
=
0
, then
a
⃗
⋅
b
⃗
=
0
\vec{a} \cdot \vec{b}=0
a
⋅
b
=
0
. But the converse need not be true. Justify your answer with an example.
Q15
EXERCISE 10.3
If the vertices
A
,
B
,
C
\mathrm{A}, \mathrm{B}, \mathrm{C}
A
,
B
,
C
of a triangle ABC are
(
1
,
2
,
3
)
,
(
−
1
,
0
,
0
)
,
(
0
,
1
,
2
)
(1,2,3),(-1,0,0),(0,1,2)
(
1
,
2
,
3
)
,
(
−
1
,
0
,
0
)
,
(
0
,
1
,
2
)
, respectively, then find
∠
A
B
C
\angle \mathrm{ABC}
∠
ABC
. [
∠
A
B
C
\angle \mathrm{ABC}
∠
ABC
is the angle between the vectors
B
A
→
\overrightarrow{\mathrm{BA}}
BA
and
B
C
→
\overrightarrow{\mathrm{BC}}
BC
].
Q16
EXERCISE 10.3
Show that the points
A
(
1
,
2
,
7
)
,
B
(
2
,
6
,
3
)
\mathrm{A}(1,2,7), \mathrm{B}(2,6,3)
A
(
1
,
2
,
7
)
,
B
(
2
,
6
,
3
)
and
C
(
3
,
10
,
−
1
)
\mathrm{C}(3,10,-1)
C
(
3
,
10
,
−
1
)
are collinear.
Q17
EXERCISE 10.3
Show that the vectors
2
i
^
−
j
^
+
k
^
,
i
^
−
3
j
^
−
5
k
^
2 \hat{i}-\hat{j}+\hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}
2
i
^
−
j
^
+
k
^
,
i
^
−
3
j
^
−
5
k
^
and
3
i
^
−
4
j
^
−
4
k
^
3 \hat{i}-4 \hat{j}-4 \hat{k}
3
i
^
−
4
j
^
−
4
k
^
form the vertices of a right angled triangle.
Q18
EXERCISE 10.3
If
a
⃗
\vec{a}
a
is a nonzero vector of magnitude '
a
a
a
' and
λ
\lambda
λ
a nonzero scalar, then
λ
a
⃗
\lambda \vec{a}
λ
a
is unit vector if
(A)
λ
=
1
\lambda=1
λ
=
1
(B)
λ
=
−
1
\lambda=-1
λ
=
−
1
(C)
a
=
∣
λ
∣
a=|\lambda|
a
=
∣
λ
∣
(D)
a
=
1
/
∣
λ
∣
a=1 /|\lambda|
a
=
1/∣
λ
∣
Q1
EXERCISE 10.4
Find
∣
a
⃗
×
b
⃗
∣
|\vec{a} \times \vec{b}|
∣
a
×
b
∣
, if
a
⃗
=
i
^
−
7
j
^
+
7
k
^
\vec{a}=\hat{i}-7 \hat{j}+7 \hat{k}
a
=
i
^
−
7
j
^
+
7
k
^
and
b
⃗
=
3
i
^
−
2
j
^
+
2
k
^
\vec{b}=3 \hat{i}-2 \hat{j}+2 \hat{k}
b
=
3
i
^
−
2
j
^
+
2
k
^
.
Q2
EXERCISE 10.4
Find a unit vector perpendicular to each of the vector
a
⃗
+
b
⃗
\vec{a}+\vec{b}
a
+
b
and
a
⃗
−
b
⃗
\vec{a}-\vec{b}
a
−
b
, where
a
⃗
=
3
i
^
+
2
j
^
+
2
k
^
\vec{a}=3 \hat{i}+2 \hat{j}+2 \hat{k}
a
=
3
i
^
+
2
j
^
+
2
k
^
and
b
⃗
=
i
^
+
2
j
^
−
2
k
^
\vec{b}=\hat{i}+2 \hat{j}-2 \hat{k}
b
=
i
^
+
2
j
^
−
2
k
^
.
Q3
EXERCISE 10.4
If a unit vector
a
⃗
\vec{a}
a
makes angles
π
3
\frac{\pi}{3}
3
π
with
i
^
,
π
4
\hat{i}, \frac{\pi}{4}
i
^
,
4
π
with
j
^
\hat{j}
j
^
and an acute angle
θ
\theta
θ
with
k
^
\hat{k}
k
^
, then find
θ
\theta
θ
and hence, the components of
a
⃗
\vec{a}
a
.
Q4
EXERCISE 10.4
Show that
(
a
⃗
−
b
⃗
)
×
(
a
⃗
+
b
⃗
)
=
2
(
a
⃗
×
b
⃗
)
(\vec{a}-\vec{b}) \times(\vec{a}+\vec{b})=2(\vec{a} \times \vec{b})
(
a
−
b
)
×
(
a
+
b
)
=
2
(
a
×
b
)
Q5
EXERCISE 10.4
Find
λ
\lambda
λ
and
μ
\mu
μ
if
(
2
i
^
+
6
j
^
+
27
k
^
)
×
(
i
^
+
λ
j
^
+
μ
k
^
)
=
0
→
(2 \hat{i}+6 \hat{j}+27 \hat{k}) \times(\hat{i}+\lambda \hat{j}+\mu \hat{k})=\overrightarrow{0}
(
2
i
^
+
6
j
^
+
27
k
^
)
×
(
i
^
+
λ
j
^
+
μ
k
^
)
=
0
.
Q6
EXERCISE 10.4
Given that
a
⃗
⋅
b
⃗
=
0
\vec{a} \cdot \vec{b}=0
a
⋅
b
=
0
and
a
⃗
×
b
⃗
=
0
→
\vec{a} \times \vec{b}=\overrightarrow{0}
a
×
b
=
0
. What can you conclude about the vectors
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
?
Q7
EXERCISE 10.4
Let the vectors
a
⃗
,
b
⃗
,
c
⃗
\vec{a}, \vec{b}, \vec{c}
a
,
b
,
c
be given as
a
1
i
^
+
a
2
j
^
+
a
3
k
^
,
b
1
i
^
+
b
2
j
^
+
b
3
k
^
a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}, b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k}
a
1
i
^
+
a
2
j
^
+
a
3
k
^
,
b
1
i
^
+
b
2
j
^
+
b
3
k
^
,
c
1
i
^
+
c
2
j
^
+
c
3
k
^
c_{1} \hat{i}+c_{2} \hat{j}+c_{3} \hat{k}
c
1
i
^
+
c
2
j
^
+
c
3
k
^
. Then show that
a
⃗
×
(
b
⃗
+
c
⃗
)
=
a
⃗
×
b
⃗
+
a
⃗
×
c
⃗
\vec{a} \times(\vec{b}+\vec{c})=\vec{a} \times \vec{b}+\vec{a} \times \vec{c}
a
×
(
b
+
c
)
=
a
×
b
+
a
×
c
.
Q8
EXERCISE 10.4
If either
a
⃗
=
0
→
\vec{a}=\overrightarrow{0}
a
=
0
or
b
⃗
=
0
→
\vec{b}=\overrightarrow{0}
b
=
0
, then
a
⃗
×
b
⃗
=
0
→
\vec{a} \times \vec{b}=\overrightarrow{0}
a
×
b
=
0
. Is the converse true? Justify your answer with an example.
Q9
EXERCISE 10.4
Find the area of the triangle with vertices
A
(
1
,
1
,
2
)
,
B
(
2
,
3
,
5
)
\mathrm{A}(1,1,2), \mathrm{B}(2,3,5)
A
(
1
,
1
,
2
)
,
B
(
2
,
3
,
5
)
and
C
(
1
,
5
,
5
)
\mathrm{C}(1,5,5)
C
(
1
,
5
,
5
)
.
Q10
EXERCISE 10.4
Find the area of the parallelogram whose adjacent sides are determined by the vectors
a
⃗
=
i
^
−
j
^
+
3
k
^
\vec{a}=\hat{i}-\hat{j}+3 \hat{k}
a
=
i
^
−
j
^
+
3
k
^
and
b
⃗
=
2
i
^
−
7
j
^
+
k
^
\vec{b}=2 \hat{i}-7 \hat{j}+\hat{k}
b
=
2
i
^
−
7
j
^
+
k
^
.
Q11
EXERCISE 10.4
Let the vectors
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
be such that
∣
a
⃗
∣
=
3
|\vec{a}|=3
∣
a
∣
=
3
and
∣
b
⃗
∣
=
2
3
|\vec{b}|=\frac{\sqrt{2}}{3}
∣
b
∣
=
3
2
, then
a
⃗
×
b
⃗
\vec{a} \times \vec{b}
a
×
b
is a unit vector, if the angle between
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
is
(A)
π
/
6
\pi / 6
π
/6
(B)
π
/
4
\pi / 4
π
/4
(C)
π
/
3
\pi / 3
π
/3
(D)
π
/
2
\pi / 2
π
/2
Q12
EXERCISE 10.4
Area of a rectangle having vertices
A
,
B
,
C
\mathrm{A}, \mathrm{B}, \mathrm{C}
A
,
B
,
C
and D with position vectors
−
i
^
+
1
2
j
^
+
4
k
^
,
i
^
+
1
2
j
^
+
4
k
^
,
i
^
−
1
2
j
^
+
4
k
^
-\hat{i}+\frac{1}{2} \hat{j}+4 \hat{k}, \hat{i}+\frac{1}{2} \hat{j}+4 \hat{k}, \hat{i}-\frac{1}{2} \hat{j}+4 \hat{k}
−
i
^
+
2
1
j
^
+
4
k
^
,
i
^
+
2
1
j
^
+
4
k
^
,
i
^
−
2
1
j
^
+
4
k
^
and
−
i
^
−
1
2
j
^
+
4
k
^
-\hat{i}-\frac{1}{2} \hat{j}+4 \hat{k}
−
i
^
−
2
1
j
^
+
4
k
^
, respectively is
(A)
1
2
\frac{1}{2}
2
1
(B) 1
(C) 2
(D) 4
Q1
Miscellaneous Exercise on Chapter 10
Write down a unit vector in XY-plane, making an angle of
30
∘
30^{\circ}
3
0
∘
with the positive direction of
x
x
x
-axis.
Q2
Miscellaneous Exercise on Chapter 10
Find the scalar components and magnitude of the vector joining the points
P
(
x
1
,
y
1
,
z
1
)
\mathrm{P}\left(x_{1}, y_{1}, z_{1}\right)
P
(
x
1
,
y
1
,
z
1
)
and
Q
(
x
2
,
y
2
,
z
2
)
\mathrm{Q}\left(x_{2}, y_{2}, z_{2}\right)
Q
(
x
2
,
y
2
,
z
2
)
.
Q3
Miscellaneous Exercise on Chapter 10
A girl walks 4 km towards west, then she walks 3 km in a direction
30
∘
30^{\circ}
3
0
∘
east of north and stops. Determine the girl's displacement from her initial point of departure.
Q4
Miscellaneous Exercise on Chapter 10
If
a
⃗
=
b
⃗
+
c
⃗
\vec{a}=\vec{b}+\vec{c}
a
=
b
+
c
, then is it true that
∣
a
⃗
∣
=
∣
b
⃗
∣
+
∣
c
⃗
∣
|\vec{a}|=|\vec{b}|+|\vec{c}|
∣
a
∣
=
∣
b
∣
+
∣
c
∣
? Justify your answer.
Q5
Miscellaneous Exercise on Chapter 10
Find the value of
x
x
x
for which
x
(
i
^
+
j
^
+
k
^
)
x(\hat{i}+\hat{j}+\hat{k})
x
(
i
^
+
j
^
+
k
^
)
is a unit vector.
Q6
Miscellaneous Exercise on Chapter 10
Find a vector of magnitude 5 units, and parallel to the resultant of the vectors
a
⃗
=
2
i
^
+
3
j
^
−
k
^
\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}
a
=
2
i
^
+
3
j
^
−
k
^
and
b
⃗
=
i
^
−
2
j
^
+
k
^
\vec{b}=\hat{i}-2 \hat{j}+\hat{k}
b
=
i
^
−
2
j
^
+
k
^
.
Q7
Miscellaneous Exercise on Chapter 10
If
a
⃗
=
i
^
+
j
^
+
k
^
,
b
⃗
=
2
i
^
−
j
^
+
3
k
^
\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}
a
=
i
^
+
j
^
+
k
^
,
b
=
2
i
^
−
j
^
+
3
k
^
and
c
⃗
=
i
^
−
2
j
^
+
k
^
\vec{c}=\hat{i}-2 \hat{j}+\hat{k}
c
=
i
^
−
2
j
^
+
k
^
, find a unit vector parallel to the vector
2
a
⃗
−
b
⃗
+
3
c
⃗
2 \vec{a}-\vec{b}+3 \vec{c}
2
a
−
b
+
3
c
.
Q8
Miscellaneous Exercise on Chapter 10
Show that the points
A
(
1
,
−
2
,
−
8
)
,
B
(
5
,
0
,
−
2
)
\mathrm{A}(1,-2,-8), \mathrm{B}(5,0,-2)
A
(
1
,
−
2
,
−
8
)
,
B
(
5
,
0
,
−
2
)
and
C
(
11
,
3
,
7
)
\mathrm{C}(11,3,7)
C
(
11
,
3
,
7
)
are collinear, and find the ratio in which B divides AC.
Q9
Miscellaneous Exercise on Chapter 10
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are (
2
a
⃗
+
b
⃗
2 \vec{a}+\vec{b}
2
a
+
b
) and (
a
⃗
−
3
b
⃗
\vec{a}-3 \vec{b}
a
−
3
b
) externally in the ratio
1
:
2
1: 2
1
:
2
. Also, show that P is the mid point of the line segment RQ .
Q10
Miscellaneous Exercise on Chapter 10
The two adjacent sides of a parallelogram are
2
i
^
−
4
j
^
+
5
k
^
2 \hat{i}-4 \hat{j}+5 \hat{k}
2
i
^
−
4
j
^
+
5
k
^
and
i
^
−
2
j
^
−
3
k
^
\hat{i}-2 \hat{j}-3 \hat{k}
i
^
−
2
j
^
−
3
k
^
. Find the unit vector parallel to its diagonal. Also, find its area.
Q11
Miscellaneous Exercise on Chapter 10
Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are
±
(
1
3
,
1
3
,
1
3
)
\pm\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)
±
(
3
1
,
3
1
,
3
1
)
.
Q12
Miscellaneous Exercise on Chapter 10
Let
a
⃗
=
i
^
+
4
j
^
+
2
k
^
,
b
⃗
=
3
i
^
−
2
j
^
+
7
k
^
\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}
a
=
i
^
+
4
j
^
+
2
k
^
,
b
=
3
i
^
−
2
j
^
+
7
k
^
and
c
⃗
=
2
i
^
−
j
^
+
4
k
^
\vec{c}=2 \hat{i}-\hat{j}+4 \hat{k}
c
=
2
i
^
−
j
^
+
4
k
^
. Find a vector
d
⃗
\vec{d}
d
which is perpendicular to both
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
, and
c
⃗
⋅
d
⃗
=
15
\vec{c} \cdot \vec{d}=15
c
⋅
d
=
15
.
Q13
Miscellaneous Exercise on Chapter 10
The scalar product of the vector
i
^
+
j
^
+
k
^
\hat{i}+\hat{j}+\hat{k}
i
^
+
j
^
+
k
^
with a unit vector along the sum of vectors
2
i
^
+
4
j
^
−
5
k
^
2 \hat{i}+4 \hat{j}-5 \hat{k}
2
i
^
+
4
j
^
−
5
k
^
and
λ
i
^
+
2
j
^
+
3
k
^
\lambda \hat{i}+2 \hat{j}+3 \hat{k}
λ
i
^
+
2
j
^
+
3
k
^
is equal to one. Find the value of
λ
\lambda
λ
.
Q14
Miscellaneous Exercise on Chapter 10
If
a
⃗
,
b
⃗
,
c
⃗
\vec{a}, \vec{b}, \vec{c}
a
,
b
,
c
are mutually perpendicular vectors of equal magnitudes, show that the vector
a
⃗
+
b
⃗
+
c
⃗
\vec{a}+\vec{b}+\vec{c}
a
+
b
+
c
is equally inclined to
a
⃗
,
b
⃗
\vec{a}, \vec{b}
a
,
b
and
c
⃗
\vec{c}
c
.
Q15
Miscellaneous Exercise on Chapter 10
Prove that
(
a
⃗
+
b
⃗
)
⋅
(
a
⃗
+
b
⃗
)
=
∣
a
⃗
∣
2
+
∣
b
⃗
∣
2
(\vec{a}+\vec{b}) \cdot(\vec{a}+\vec{b})=|\vec{a}|^{2}+|\vec{b}|^{2}
(
a
+
b
)
⋅
(
a
+
b
)
=
∣
a
∣
2
+
∣
b
∣
2
, if and only if
a
⃗
,
b
⃗
\vec{a}, \vec{b}
a
,
b
are perpendicular, given
a
⃗
≠
0
→
,
b
⃗
≠
0
→
\vec{a} \neq \overrightarrow{0}, \vec{b} \neq \overrightarrow{0}
a
=
0
,
b
=
0
.
Q16
Miscellaneous Exercise on Chapter 10
If
θ
\theta
θ
is the angle between two vectors
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
, then
a
⃗
⋅
b
⃗
≥
0
\vec{a} \cdot \vec{b} \geq 0
a
⋅
b
≥
0
only when
(A)
0
<
θ
<
π
2
0<\theta<\frac{\pi}{2}
0
<
θ
<
2
π
(B)
0
≤
θ
≤
π
2
0 \leq \theta \leq \frac{\pi}{2}
0
≤
θ
≤
2
π
(C)
0
<
θ
<
π
0<\theta<\pi
0
<
θ
<
π
(D)
0
≤
θ
≤
π
0 \leq \theta \leq \pi
0
≤
θ
≤
π
Q17
Miscellaneous Exercise on Chapter 10
Let
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
be two unit vectors and
θ
\theta
θ
is the angle between them. Then
a
⃗
+
b
⃗
\vec{a}+\vec{b}
a
+
b
is a unit vector if
(A)
θ
=
π
4
\theta=\frac{\pi}{4}
θ
=
4
π
(B)
θ
=
π
3
\theta=\frac{\pi}{3}
θ
=
3
π
(C)
θ
=
π
2
\theta=\frac{\pi}{2}
θ
=
2
π
(D)
θ
=
2
π
3
\theta=\frac{2 \pi}{3}
θ
=
3
2
π
Q18
Miscellaneous Exercise on Chapter 10
The value of
i
^
⋅
(
j
^
×
k
^
)
+
j
^
⋅
(
i
^
×
k
^
)
+
k
^
⋅
(
i
^
×
j
^
)
\hat{i} \cdot(\hat{j} \times \hat{k})+\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{k} \cdot(\hat{i} \times \hat{j})
i
^
⋅
(
j
^
×
k
^
)
+
j
^
⋅
(
i
^
×
k
^
)
+
k
^
⋅
(
i
^
×
j
^
)
is
(A) 0
(B) -1
(C) 1
(D) 3
Q19
Miscellaneous Exercise on Chapter 10
If
θ
\theta
θ
is the angle between any two vectors
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
, then
∣
a
⃗
⋅
b
⃗
∣
=
∣
a
⃗
×
b
⃗
∣
|\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|
∣
a
⋅
b
∣
=
∣
a
×
b
∣
when
θ
\theta
θ
is equal to
(A) 0
(B)
π
4
\frac{\pi}{4}
4
π
(C)
π
2
\frac{\pi}{2}
2
π
(D)
π
\pi
π
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