The document contains educational material focused on scalars and vectors, including illustrative examples, theory questions, textbook problems, practice problems, and multiple-choice questions. It discusses vector addition, components, scalar and vector products, and various properties and laws related to vectors. The content is structured to support learning and problem-solving in physics, particularly in the context of vectors.

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F.Y.J.C. Scalars and Vectors
Scalars and Vectors
❖ Illustrative Examples
1) The vectors and have magnitude 3 units and 4 units respectively. is the
resultant of and . Find the magnitude and direction of when the angle
between and 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. is the required vector.]
4) Find m if and have the same directions. (Ans. )
5) If and Find the angle between and
(Ans. )
6) The angular momentum is a vector product of position vector and linear
momentum If and L. (Ans. )
7) Find the area o1'the parallelogram with adjacent sides formed by and .
where and expressed in meter. (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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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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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.
❖ Textbook Problems
1) The forces and of magnitude 5N each inclined to each other at act on
body. Find the resultant force acting on the body. (Ans : F = 8. 662N)
2) A force 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. (Ans : 41J)
3) If 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 and its magnitude.

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F.Y.J.C. Scalars and Vectors
(Ans : units)
5) Find ‘a’ if and are perpendicular to one another.
(Ans : a = 8/3)
6) If and find (i) (ii)
(Ans : (i) (ii) )
7) Fins the vector that should be added to the sum of and to
given a unit vector along the X-axis. (Ans : )
8) Find unit vectors perpendicular to the plane of the vectors, and
(Ans : )
9) Find the area of a triangle formed by and as adjacent
sides measured in metre. (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 and find the magnitude and direction of
i. ii. iii. iv.

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(Ans. i. 5, ii. 10, iii. iv.
v. )
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 . Find the vector component of the velocity
along the line and its magnitude. (Ans. )
8. A velocity of 10 ms -1 has its Y-component Calculate the X-component.
If find the angle between and . (Ans. 900)
➢ Type-II : problems based on product of vectors
9. Find the angle between the vectors and . (Ans. 600)
10. If and are two vectors, find (Ans. 5.91)
11. and are unit vectors along X-axis and Y-axis respectively. What is the magnitude
and direction of the vector and What are the components of a vector
along the directions of and ?
12. Find where and (Ans. 3)
13. Find the angle between the vectors and (Ans. 900)
14. If and then find the value of
(Ans. )

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15. The diagonals of a parallelogram are given by the vectors and .
Find the area of the parallelogram. (Ans. 8.66 sq. unit)
16. If and determine the unit vector parallel to . Also find
the sine of the angle between and . (Ans. )
➢ Type-III : Miscellaneous
17. If and , find a vector having the same magnitude as and parallel
to . (Ans. )
18. Determine the work done when a force acting on a particle produces a
displacement m. (Ans.15 J)
19. If and 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. (Ans. unit, 10 unit)
23. If vectors and 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)

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❖ Multiple choice Questions
1. Which of the following is a vector?
(A) Speed (B) Displacement (C) mass (D) time
2. The angle between the vectors and is
(A) 900 (B) 600 (C) 300 (D) 00
3. Two quantities of 5 and 12 unit when added gives a quantity 13 unit. This quantity
is
(A) Time (B) Mass (C) Linear momentum (D) speed
4. If and a and b are non zero vectors, then
(A) and are antiparallel (C) and are zero parallel vectors.
(B) (D) is to .
5. The equation + = is
(A) Meaningless (C) may be possible for limited values of 'a'
(B) always true (D) true only when = 0
6. The resultant of two vectors of magnitude is also . They act at an angle
(A) 600 (B) 900 (C) 1200 (D) 1800
7. Given that which of the following relations is necessarily valid ?
(A) P < Q (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) (B) (C) (D)

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11. The vectors and are such that + = and A2 + B2 = C2. Angle between
positive directions of and is
(A) (B) 0 (C) (D)
12. If = + and magnitudes of , and are 5, 4 and 3 unit respectively, then
angle between and is
(A) Sin-1(3/4) (B) Cos-1(5/3) (C) tan-1(5/3) (D) cos-1(3/5)
13. The vector projection of a vector 8 + 6 on Y-axis is
(A) 10 (B) 0 (C) 8 (D) 6
14. The expression, is a
(A) unit vector (C) vector of magnitude
(B) null vector (D) scalar
15. What is the angle between and ?
(A) 00 (B) (C) (D) None of the above
16. is a unit vector along X-axis. If then is
(A) (B) (C) (D)
17. The magnitude of scalar product of the vectors and is
(A) 20 (B) 22 (C) 26 (D) 29
18. If and , 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 satisfy the relation and then is parallel to
(A) (B) (C) (D) .

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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) (D)
21. The maximum value of magnitude of is
(A) A – B (B) A (C) A + B (D)
22. The magnitude of the X and Y components of are 7 and 6. Also the magnitudes of
the X and Y components of are 11 and 9 respectively. What is the magnitude
of ?
(A) 5 (B) 6 (C) 8 (D) 9
23. If and then component of along is
(A) (B) (C) (D)
24. If then vector 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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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 (D) 1200
31. Let the angle between two non-zero vectors and be 1200 and its resultant be C
then
(A) must be equal to (C) must be greater than
(B) must be less than (D) must be zero.
32. If is the unit vector in the direction of then
(A) (B) (C) (D)
33. What is the maximum number of components into which a force can be resolved?
(A) Two (B) Three (C) Four (D) Any number
34. The unit vector along is
(A) (B) (C) (D)
35. If vectors and are perpendicular to each other, then
(A) (B) (C) (D)
36. If and are non-zero vectors and and then magnitude of is
(A) A + C (B) AC (C) AB (D) BC
❖ Answers of Multiple choice Questions
1.B 2.A 3.C 4.D 5.D 6.C 7.D
8.C 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
36.B