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Scalars and Vectors ❖ Illustrative Examples 1) The vectors
and
have magnitude 3 units and 4 units respectively.
resultant of
and
. Find the magnitude and direction of
between
and
is the
when the angle
is 30o.
Data : P = 3 unit, Q = 4 unit,
= 30o
(Ans.
)
2) Two points A and B in space have the co-ordinates (2,-l,3) and (4,2,5) respectively. Find vector AB.
(Ans.
)] 3) What vector added to
and
will give a unit vector along negative y
– axis?
(Ans.
4) Find m if 5) If
and and
is the required vector.]
have the same directions. Find the angle between
(Ans.
and (Ans.
6) The angular momentum momentum
is a vector product of position vector
If
and
L.
and
)
and linear
(Ans.
7) Find the area o1'the parallelogram with adjacent sides formed by where
)
expressed in meter.
) and
.
(Ans.
)
❖ Theory Questions 1)
Define scalars and vectors. Give two examples of each.
2)
What is equal vector and null vector?
3)
What is triangle law of vector addition?
4)
What is law of polygon of vectors?
5)
State and prove parallelogram law of vector addition and determine magnitude and direction of resultant vector.
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CONTACT: 9821131002/9029004242
6)
Define unit vector and give its physical significance.
7)
Explain what is resolution of vectors.
8)
What are rectangular components of vectors? Explain their use.
9)
Define scalar product of two vectors. State the characteristics of scalar product.
10) Derive the expression for scalar product of two vectors in terms of their scalar components. 11) Define vector product of two vectors. State the characteristics of vector product. 12) Derive an expression for cross product of two vectors and express it in determine form. 13) Show that magnitude of vector product of two vectors is numerically equal to the area of a parallelogram formed by the two vectors. 14) Distinguish between dot product and cross product. 15) Distinguish Between Scalars and Vectors. 16) Explain representation of a vector graphically and symbolically. 17) A vector has both magnitude and direction Does it mean that anything that has magnitude and direction is necessarily a vector? 18) Define the terms: i.
Parallel vectors
ii. Antiparallel vectors iii. Collinear vector. iv.
Negative vectors
19) Define the terms. i. Position vector 20) Define the following terms. i. Resultant vector ii. Composition of vectors. 21) Explain addition of vectors. 22) Using triangle law of vector addition explain the process of adding two vectors which are not lying in a straight line. 23) Explain, how two vectors are subtracted to find their resultant by using triangle law of vector addition. 24) Prove that: addition of two vectors obey commutative law. 25) Prove the associative property of vector addition. 26) Explain rectangular unit vectors. 27) Define Components of a vector. 28) Whether it is possible to add two vectors representing physical quantities having different dimensions?
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INFOMATICA ACADEMY
CONTACT: 9821131002/9029004242
29) Is it possible to add two velocities using triangle law? 30) Whether the subtraction of given vectors is commutative or associative? 31) The diagonal of the parallelogram made by two vectors as adjacent sides is not passing through common point of two vectors. What does it represent ? 32) 1f
then what can be the angle between
and
?
33) What are (i) dimensions and (ii) units of a unit vector? 34) If the frame of reference is rotated or displaced, then what happens to the vector and its components? 35) Define scalar product of two vectors. Give suitable examples. 36) Give the geometrical meaning of dot product of two vectors. 37) Define vector product of two vectors. Explain vector product of two vectors with suitable examples. 38) State right handed screw rule. 39) State with reasons whether the following algebraic operations with scalar and vector physical quantities are meaningful. a. Adding any two scalars, b. Adding a scalar to a vector of the same dimensions, c. Multiplying any vector by any scalar, d. Multiplying any two scalars, e. Adding any two vectors.
â?– Textbook Problems 1) The forces
and
of magnitude 5N each inclined to each other at
body. Find the resultant force acting on the body. 2) A force
act on
(Ans : F = 8. 662N)
acting on a particle produces a displacements of where F is expressed in newton and S in meter. Find the work done
by the force. 3) If
(Ans : 41J) and
find the component of
along (Ans :
4) In a Cartesian co-ordinates system, the co-ordinates of two points O and Q are (2, 4, 4) and (-2, -3, 7) respectively. Find
F.Y.J.C.
and its magnitude.
Scalars and Vectors
)
INFOMATICA ACADEMY
CONTACT: 9821131002/9029004242 (Ans :
5) Find ‘a’ if
and
units)
are perpendicular to one another. (Ans : a = 8/3)
6) If
and
find (i)
(ii) (Ans : (i)
7) Fins the vector that should be added to the sum of
(ii)
)
and
given a unit vector along the X-axis.
to (Ans :
8) Find unit vectors perpendicular to the plane of the vectors,
)
and
(Ans : 9) Find the area of a triangle formed by
)
and
sides measured in metre.
as adjacent (Ans : 6.10 m2)
❖ Practice Problems ➢ Type-I : problems based on addition and subtraction of vectors. 1. Find the vector drawn from the point (-4, 10, 7) to the point (3, -2,l). Also find its magnitude.
(Ans.
)
2. Find unit vector parallel to the resultant of the vectors
and
(Ans.
.
)
3. One of the rectangular components of a velocity of 80kmh-1 is 40 kmh-l. Find the other component.
(Ans. 69.28 Kmh-1)
4. Determine the angles which the vector
makes with X,Y and Z axes.
(Ans. The vectors makes angles of 450,900 and 450 w.r.t. X,Y,Z respectively) 5. If i.
F.Y.J.C.
and find the magnitude and direction of ii.
iii.
iv.
Scalars and Vectors
INFOMATICA ACADEMY (Ans. i. 5,
CONTACT: 9821131002/9029004242
ii. 10,
v.
iii.
iv.
)
6. Points, P and Q, have co-ordinates (1, 2, 3) and (4, 5, 6), respectively. Find
.
(Ans.
)
7. Find the angle between two equal vectors if their resultant is equal to either of them. (Ans. 1200) The velocity of a particles is
along the line
. Find the vector component of the velocity
and its magnitude.
(Ans.
8. A velocity of 10 ms -1 has its Y-component If
find the angle between
and
)
Calculate the X-component. .
(Ans. 900)
➢ Type-II : problems based on product of vectors 9. Find the angle between the vectors 10. If 11.
and and
are two vectors, find
and
along the directions of where
(Ans. 600) (Ans. 5.91)
What are the components of a vector and
?
and
13. Find the angle between the vectors
14. If
.
are unit vectors along X-axis and Y-axis respectively. What is the magnitude
and direction of the vector
12. Find
and
and
(Ans. 3) and
(Ans. 900)
then find the value of (Ans.
F.Y.J.C.
Scalars and Vectors
)
INFOMATICA ACADEMY
CONTACT: 9821131002/9029004242
15. The diagonals of a parallelogram are given by the vectors
and
Find the area of the parallelogram. 16. If
and
.
(Ans. 8.66 sq. unit)
determine the unit vector parallel to
the sine of the angle between
and
.
. Also find
(Ans.
)
➢ Type-III : Miscellaneous 17. If to
and
, find a vector having the same magnitude as
.
and parallel (Ans.
18. Determine the work done when a force displacement 19. If
acting on a particle produces a
m.
and
)
(Ans.15 J)
find the unit vector Parallel to
. Also find the vector
Perpendicular to p and Q of magnitude 6 units.
(Ans. i.
ii.
)
20. Find the area of the triangle formed by the tips of the vectors And
.
(Ans. 6.4 sq. units)
21. Rain is falling vertically with a speed of 35 m/s. wind starts blowing at a speed of 12 m/s in east to west direction. In which direction should a boy waiting at a bus stop hold his umbrella? (NCERT) .
(Ans.190 )
22. Find the components along X and Y axes of a vector 20 unit long when it makes an angle of 30o with the X-axis. 23. If vectors
and
(Ans.
unit, 10 unit)
have magnitudes 5, 12 and 13 unit respectively and
find the angle between
and
.
(Ans.
24. A body acted upon by a force of 50 N is displaced through a distance of 10 m in a direction making an angle of 60o with force. Calculate the work done by the force . (Ans. 250 J)
F.Y.J.C.
Scalars and Vectors
)
INFOMATICA ACADEMY
CONTACT: 9821131002/9029004242
â?– Multiple choice Questions 1. Which of the following is a vector? (A) Speed
(B) Displacement
(C) mass
2. The angle between the vectors (A) 900
and
(B) 600
(D) time is
(C) 300
(D) 00
3. Two quantities of 5 and 12 unit when added gives a quantity 13 unit. This quantity is (A) Time 4. If (A)
(B) Mass
(C) Linear momentum
(D) speed
and a and b are non zero vectors, then and
are antiparallel
(B) 5. The equation
+ =
(C)
and
(D)
is
are zero parallel vectors. to
.
is
(A) Meaningless
(C) may be possible for limited values of 'a'
(B) always true
(D) true only when
6. The resultant of two vectors of magnitude (A) 600 7. Given that (A) P < Q
(B) 900
is also
(C) 1200
=0
. They act at an angle (D) 1800
which of the following relations is necessarily valid ? (B) P > Q
(C) P = Q
(D) None of these
8. The minimum number of numerically equal vectors whose vector sum can be zero is (A) 4
(B) 3
(C) 2
(D) 1
9. A force of 60 N acting perpendicular to a force of 80 N, magnitude of resultant force is (A) 20 N
(B) 70 N
(C) 100 N
(D) 140 N
10. A river is flowing at the rate of 6 km h-1. A man swims across it with a velocity of 9 km h-1. The resultant velocity of the man will be (A)
F.Y.J.C.
(B)
(C)
(D)
Scalars and Vectors
INFOMATICA ACADEMY 11. The vectors
and
are such that
positive directions of (A) 12. If
CONTACT: 9821131002/9029004242
and
+
+
(C)
and magnitudes of
angle between (A) Sin-1(3/4)
and
,
(D)
and
are 5, 4 and 3 unit respectively, then
(C) tan-1(5/3)
13. The vector projection of a vector 8
+6
(B) 0
(D) cos-1(3/5)
on Y-axis is
(C) 8
14. The expression,
(D) 6
is a
(A) unit vector
(C) vector of magnitude
(B) null vector
(D) scalar
15. What is the angle between
(A) 00 16.
and
(B)
?
(C)
(D) None of the above
is a unit vector along X-axis. If (A)
(B)
then
(C)
18. If
(B) 22 and
is (D)
17. The magnitude of scalar product of the vectors (A) 20
between
is
(B) Cos-1(5/3)
(A) 10
and A2 + B2 = C2. Angle
is
(B) 0 =
=
and
(C) 26
is
(D) 29
, then the area of parallelogram formed from these
vectors as the adjacent sides will be (A) 2
square units
(C) 6
square units
(B) 4
square units
(D) 8
square units.
19. Three vectors (A)
F.Y.J.C.
satisfy the relation (B)
(C)
and
then (D)
is parallel to
.
Scalars and Vectors
INFOMATICA ACADEMY
CONTACT: 9821131002/9029004242
20. What vector must be added to the sum of two vectors
and
so that
the resultant is a unit vector along Z axis ? (A)
(B)
(C)
21. The maximum value of magnitude of (A) A â&#x20AC;&#x201C; B
(B) A
(D)
is (C) A + B
(D)
22. The magnitude of the X and Y components of the X and Y components of of
are 7 and 6. Also the magnitudes of
are 11 and 9 respectively. What is the magnitude
?
(A) 5
(B) 6
23. If
and
(A) 24. If
(C) 8
(D) 9
then component of
(B) then vector
(C)
along
is
(D)
must be
(A) zero vector
(C) Non zero vector
(B) unit vector
(D) equal to
25. Choose the WRONG statement (A) The division of vector by scalar is valid. (B) The multiplication of vector by scalar is va1id. (C) The multiplication of vector by another vector is valid by using vector algebra. (D) The division of a vector by another vector is valid by using vector algebra. 26. A person moves from a point S and walks along the path which is a square of each side 50 m. He runs east, south, then west and finally north. Then the total displacement covered is (A) 200 m
(B) 100m
(C) 50
m
(D) Zero
27. Walking on an inclined plane, the road is an example of (A) Vectors
(B) scalars
(C) resolution of Vectors
(D) null vector
28. Which of the following is a scalar? (A) Electric field
(C) Angular frequency
(B) Angular momentum
(D) Torque.
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INFOMATICA ACADEMY
CONTACT: 9821131002/9029004242
29. Which of the following is a vector? (A) Pressure
(C) Angle
(B) Gravitational potential
(D) Current density
30. The resultant of two forces of 3 N and 4 N is 5 N, the angle between the forces is (A) 300
(B) 600
(C) 900
31. Let the angle between two non-zero vectors
(D) 1200
and
be 1200 and its resultant be C
then (A)
must be equal to
(B) 32. If
must be less than is the unit vector in the direction of
(A)
(B)
(C)
must be greater than
(D)
must be zero.
then
(C)
(D)
33. What is the maximum number of components into which a force can be resolved? (A) Two
(B) Three
34. The unit vector along
(A)
(C) Four
(D) Any number
(C)
(D)
is
(B)
35. If vectors and are perpendicular to each other, then (A) 36. If
(B) and
(C)
are non-zero vectors and
(A) A + C
(B) AC
(D) and
(C) AB
then magnitude of (D) BC
â?&#x2013; Answers of Multiple choice Questions 1.B
2.A
3.C
4.D
5.D
6.C
7.D
9.C
10.C
11.A
12.B
13.B
14.A
15.D
16.B
17.C
18.D
19.C
20.B
21.C
22.A
23.D
24.A
25.D
26.D
27.A
28.C
29.D
30.C
31.C
32.A
33.D
34.C
35.A
8.C
36.B
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Scalars and Vectors
is