Class 12 Maths Chapter 10.docxchapter 10 vector algebra-Exercise: 10.1
Question:1 Represent graphically a displacement of 40 km, east of north.
Answer:
Represent graphically a displacement of 40 km, east of north.
N,S,E,W are all 4 direction north,south,east,west respectively.
is displacement vector which
= 40 km.
makes an angle of 30 degree east of north as shown in figure.
Question:2 (1) Classify the following measures as scalars and vectors.
10Kg
Answer:
10kg is a scalar quantity as it has only magnitude.
Question:2 (2) Classify the following measures as scalars and vectors. 2 meters north
west
Answer:
This is a vector quantity as it has both magnitude and direction.
Question:2 (3) Classify the following measures as scalars and vectors.
Answer:
This is a scalar quantity as it has only magnitude.
Question:2 (4) Classify the following measures as scalars and vectors. 40 watt
Answer:
This is a scalar quantity as it has only magnitude.
Question:2 (5) Classify the following measures as scalars and vectors.
Answer:
This is a scalar quantity as it has only magnitude.
Question:2 (6) Classify the following measures as scalars and vectors.
Answer:
This is a Vector quantity as it has magnitude as well as direction.by looking at the unit,
we conclude that measure is acceleration which is a vector.
Question:3 Classify the following as scalar and vector quantities.
(1) time period
This is a scalar quantity as it has only magnitude.
Question:3 Classify the following as scalar and vector quantities.
(2) distance
Distance is a scalar quantity as it has only magnitude.
Question:3 Classify the following as scalar and vector quantities.
(3) force
Answer:
Force is a vector quantity as it has both magnitude as well as direction.
Question:3 Classify the following as scalar and vector quantities.
(4) velocity
Answer:
Velocity is a vector quantity as it has both magnitude and direction.
Question:3 Classify the following as scalar and vector quantities.
(5) work done
Answer:
work done is a scalar quantity, as it is the product of two vectors.
Question:4 In Fig 10.6 (a square), identify the following vectors.
(1) Coinitial
Answer:
Since vector and vector are starting from the same point, they are coinitial.
Question:4 In Fig 10.6 (a square), identify the following vectors.
(2) Equal
Answer:
Since Vector and Vector both have the same magnitude and same direction, they
are equal.
Question:4 In Fig 10.6 (a square), identify the following vectors.
(3) Collinear but not equal
Answer:
Since vector and vector have the same magnitude but different direction, they are
colinear and not equal.
(1) and are collinear.
Answer:
True, and are collinear. they both are parallel to one line hence they are colinear.
Question:5 Answer the following as true or false.
(2) Two collinear vectors are always equal in magnitude.
Answer:
False, because colinear means they are parallel to the same line but their magnitude
can be anything and hence this is a false statement.
Question:5 Answer the following as true or false.
(3) Two vectors having same magnitude are collinear.
Answer:
False, because any two non-colinear vectors can have the same magnitude.
Question:5 Answer the following as true or false.
(4) Two collinear vectors having the same magnitude are equal.
Answer:
False, because two colinear vectors with the same magnitude can have opposite
direction
NCERT solutions for class 12 maths chapter 10 vector algebra-Exercise: 10.2
Question:1 Compute the magnitude of the following vectors:
(1)
Answer:
Here
(2)
Answer:
Here,
(3)
Answer:
Here,
Answer:
The magnitude of both vector
Question:3 Write two different vectors having same direction.
Answer:
Two different vectors having the same direction are:
Question:4 Find the values of x and y so that the vectors and are
equal.
Answer:
will be equal to when their corresponding components are equal.
Hence when,
and
Question:5 Find the scalar and vector components of the vector with initial point (2, 1)
and terminal point (– 5, 7).
Answer:
Now,
Question:6 Find the sum of the vectors
Answer:
Given,
Now, The sum of the vectors:
Question:7 Find the unit vector in the direction of the vector
Answer:
Given
A unit vector in the direction of
Question:8 Find the unit vector in the direction of vector , where P and Q are the
points (1, 2, 3) and (4, 5, 6), respectively.
Answer:
Given P = (1, 2, 3) and Q = (4, 5, 6)
A vector in direction of PQ
Magnitude of PQ
Question:9 For given vectors, and , find the unit
vector in the direction of the vector .
Answer:
Given
Now,
Now a unit vector in the direction of
Question:10 Find a vector in the direction of vector which has magnitude
8 units.
A vector in direction of and whose magnitude is 8 =
Question:11 Show that the vectors and are collinear.
Answer:
Let
Hence here
Here
Question:12 Find the direction cosines of the vector
Answer:
Let
Hence direction cosine of are
Question:13 Find the direction cosines of the vector joining the points A(1, 2, –3) and
B(–1, –2, 1), directed from A to B.
Answer:
Given
Vector joining A and B Directed from A to B
Hence Direction cosines of vector AB are
Question:14 Show that the vector is equally inclined to the axes OX, OY and
OZ.
Answer:
Let
Hence direction cosines of this vectors is
Let , and be the angle made by x-axis, y-axis and z- axis respectively
Now as we know,
,
Hence Given vector is equally inclined to axis OX,OY and OZ.
Question:15 (1) Find the position vector of a point R which divides the line joining two
points P and Q whose position vectors are and respectively, in
the ratio 2 : 1 internally
Answer:
As we know
The position vector of the point R which divides the line segment PQ in ratio m:n
internally:
Here
m:n = 2:1
And Hence
Question:15 (2) Find the position vector of a point R which divides the line joining two
points P and Q whose position vectors are and respectively, in
the ratio 2 : 1 externally
Answer:
As we know
The position vector of the point R which divides the line segment PQ in ratio m:n
externally:
Here
m:n = 2:1
And Hence
Question:16 Find the position vector of the mid point of the vector joining the points
P(2, 3, 4) and Q(4, 1, –2).
Answer:
Given
Position Vector of point Q =
The position vector of R which divides PQ in half is given by:
Question:17 Show that the points A, B and C with position vectors, , respectively form
the vertices of a right angled triangle.
Answer:
Given
Now,
Hence ABC is a right angle triangle.
Question:18 In triangle ABC (Fig 10.18), which of the following is not true:
Answer:
From here
Option C is False.
Question:19 If are two collinear vectors, then which of the following are incorrect:
(A) for some saclar
(D) both the vectors have same direction, but different magnitudes.
Answer:
If two vectors are collinear then, they have same direction or are parallel or anti-
parallel.
Therefore,
They can be expressed in the form where a and b are vectors and is some
scalar quantity.
Now,
(b) is a scalar quantity so its value may be equal to
Therefore,
Therefore, (c) is not true.
D) The vectors and can have different magnitude as well as different directions.
Therefore, (d) is not true.
Therefore, the correct options are (C) and (D).
NCERT solutions for class 12 maths chapter 10 vector algebra-Exercise: 10.3
Question:1 Find the angle between two vectors with magnitudes ,
respectively having .
So,
Answer:
Hence the angle between
Question:3 Find the projection of the vector on the vector
Answer:
Let
Question:4 Find the projection of the vector on the vector
Answer:
Let
Hence, projection of vector on is
Question:5 Show that each of the given three vectors is a unit vector: Also, show that
they are mutually perpendicular to each other.
Answer:
Given
Now,
Question:6 Find , if .
Question:7 Evaluate the product .
To evaluate the product
Question:8 Find the magnitude of two vectors , having the same magnitude
and such that the angle between them is and their scalar product is 1/2
Answer:
Hence, the magnitude of two vectors
Question:9 Find , if for a unit vector
Answer:
Given in the question that
And we need to find
So the value of is
Question:10 If are such that is perpendicular to , then find the value of
Answer:
and is perpendicular to
so the value of -
As is perpendicular to
Question:11 Show that is perpendicular to , for any two nonzero
vectors .
Answer:
are two non-zero vectors
According to the question
Hence is perpendicular to .
vector ?
Answer:
Therefore is a zero vector. Hence any vector will satisfy
Question:13 If are unit vectors such that , find the value
of
Answer:
Answer- the value of is
Question:14 If either vector . But the converse need
not be true. Justify your answer with an example
Answer:
Let
Hence here converse of the given statement is not true.
Question:15 If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2),
respectively, then find is the angle between the
vectors .
Answer:
Hence angle between them ;
Answer - Angle between the vectors is
Question:16 Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear.
Answer:
Given in the question
A=(1, 2, 7), B=(2, 6, 3) and C(3, 10, –1)
To show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear
As we see that
Hence point A, B , and C are colinear.
Question:17 Show that the vectors form the vertices of a right angled triangle.
Answer:
Given the position vector of A, B , and C are
To show that the vectors form the vertices of a right angled triangle
Here we see that
Hence A,B, and C are the vertices of a right angle triangle.
Question:18 If is a nonzero vector of magnitude ‘a’ and a nonzero scalar,
then is unit vector if
Answer:
Given is a nonzero vector of magnitude ‘a’ and a nonzero scalar
is a unit vector when
Hence the correct option is D.
NCERT solutions for class 12 maths chapter 10 vector algebra-Exercise: 10.4
Question:1 Find
Now,
So the value of is
Question:2 Find a unit vector perpendicular to each of the vector ,
where
Answer:
And a unit vector in this direction :
Hence Unit vector perpendicular to each of the
vector is .
Question:3 If a unit vector makes angles with with and an acute
angle with then find and hence, the components of .
Answer:
As product of a vector with itself is always Zero,
As cross product of a and b is equal to negative of cross product of b and a.
= RHS
Question:5 Find and if
From Here we get,
Question:6 Given that and . What can you conclude about the
vectors ?
Answer:
When , either are perpendicular to each other
When either are parallel to each other
Since two vectors can never be both parallel and perpendicular at same time,we
conclude that
that
Answer:
Now,
Now
your answer with an example.
Answer:
No, the converse of the statement is not true, as there can be two non zero vectors, the
cross product of whose are zero. they are colinear vectors.
Consider an example
Hence converse of the given statement is not true.
Question:9 Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5,
5).
Answer:
Given in the question
vertices A=(1, 1, 2), B=(2, 3, 5) and C=(1, 5, 5). and we need to find the area of the
triangle
The area of the triangle is square units
Question:10 Find the area of the parallelogram whose adjacent sides are determined
by the vectors and .
Area of parallelogram with adjescent side and ,
The area of the parallelogram whose adjacent sides are determined by the
vectors and is
then is a unit vector, if the angle between is
Answer:
As given is a unit vector, which means,
Hence the angle between two vectors is . Correct option is B.
Question:12 Area of a rectangle having vertices A, B, C and D with position vectors
(A)1/2
Now,
Hence option C is correct.
NCERT solutions for class 12 maths chapter 10 vector algebra- Miscellaneous Exercise
Question:1 Write down a unit vector in XY-plane, making an angle of with the
positive direction of x-axis.
a unit vector in XY-Plane making an angle with x-axis :
Hence for
Answer- the unit vector in XY-plane, making an angle of with the positive direction
of x-axis is
Question:2 Find the scalar components and magnitude of the vector joining the points
Answer:
Given in the question
And we need to finrd the scalar components and magnitude of the vector joining the
points P and Q
Magnitiude of vector PQ
Scalar components are
Question:3 A girl walks 4 km towards west, then she walks 3 km in a direction east
of north and stops. Determine the girl’s displacement from her initial point of departure.
Answer:
Position vector =
Now as she moves 3km in direction 30 degree east of north.
hence final position vector is;
Question:4 If , then is it true that ? Justify your answer.
Answer:
the condition satisfies in the triangle.
also, in a triangle,
if then we can not conclude that
Question:5 Find the value of x for which is a unit vector.
Answer:
The value of x is
Question:6 Find a vector of magnitude 5 units, and parallel to the resultant of the
vectors
Answer:
Now, a unit vector of magnitude in direction of
Hence the required vector is
Question:7 If , find a unit vector parallel to the vector .
Answer:
Now,
OR
Question:8 Show that the points A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7) are collinear,
and find the ratio in which B divides AC.
Answer:
Given in the question,
points A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7)
now let's calculate the magnitude of the vectors
As we see that AB = BC + AC, we conclude that three points are colinear.
we can also see from here,
Point B divides AC in the ratio 2 : 3.
Question:9 Find the position vector of a point R which divides the line joining two points
P and Q whose position vectors are externally in the ratio 1: 2.
Also, show that P is the mid point of the line segment RQ.
Answer:
Given, two vectors
the point R which divides line segment PQ in ratio 1:2 is given by
Hence position vector of R is .
Now, Position vector of the midpoint of RQ
which is the position vector of Point P . Hence, P is the mid-point of RQ
Question:10 The two adjacent sides of a parallelogram
are . Find the unit vector parallel to its diagonal. Also,
find its area.
Given, two adjacent sides of the parallelogram
The diagonal will be the resultant of these two vectors. so
resultant R:
Hence unit vector along the diagonal of the parallelogram
Now,
Hence the area of the parallelogram is .
Question:11 Show that the direction cosines of a vector equally inclined to the axes
OX, OY and OZ are
Answer:
Let a vector is equally inclined to axis OX, OY and OZ.
let direction cosines of this vector be
Now
both
Answer:
Given,
Let
now, since it is given that d is perpendicular to and , we got the condition,
and
And
And
here we got 2 equation and 3 variable. one more equation will come from the condition:
so now we have three equation and three variable,
On solving this three equation we get,
,
Hence Required vector :
Question:13 The scalar product of the vector with a unit vector along the
sum of vectors and is equal to one. Find the value of .
Answer:
unit vector along
squaring both the side,
Question:14 If are mutually perpendicular vectors of equal magnitudes, show
that the vector is equally inclined to .
Answer:
Given
and
Now,
Question:15 Prove that , if and only if are
perpendicular, given
LHS=
Hence proved.
Question:16 Choose the correct answer If is the angle between two
vectors , then only when
this will satisfy when
Hence option B is the correct answer.
Question:17 Choose the correct answer. Let be two unit vectors and is the
angle between them. Then is a unit vector if
Answer:
be two unit vectors and is the angle between them
also
Hence option D is correct.
Question:18 The value of is
(A) 0
(B) –1
(C) 1
(D) 3
To find the value of
Hence option C is correct.
Question:19 Choose the correct. If is the angle between any two vectors ,
then when
To find the value of
Hence option D is correct.
NCERT solutions for class 12 maths
chapter 10 vector algebra-Exercise: 10.1
NCERT solutions for class 12 maths chapter 10 vector algebra-Exercise: 10.2
NCERT solutions for class 12 maths chapter 10 vector algebra-Exercise: 10.3
NCERT solutions for class 12 maths chapter 10 vector algebra-Exercise: 10.4
NCERT solutions for class 12 maths chapter 10 vector algebra-Miscellaneous Exercise