Class 12 NCERT Solutions- Mathematics Part ii – Chapter 10 – Vector Algebra Exercise 10.1

Question 1: Represent graphically a displacement of 40 km, 30° east of north.

Solution:

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To graphically represent a displacement of 40 km, 30° east of north:

1. Choose a Scale: For example, 1 cm = 10 km. Hence, 40 km = 4 cm.
2. Draw the Axes: Align the north direction with the positive y-axis and the east direction with the positive x-axis.
3. Determine the Angle: The displacement is 30° east of north.
4. Plot the Displacement: From the origin, use a protractor to measure 30° from the north towards the east.
5. Measure the Distance: Measure 4 cm along the 30° line from the origin.
6. Draw the Vector: Draw an arrow from the origin to the endpoint, representing the vector.

Question 2. Classify the following measures as scalars and vectors.

(i) 10 kg

(ii) 2 meters north-west

(iii) 40°

(iv) 40 watt

(v) 10–19 coulomb

(vi) 20 m/s2

Solution:

(i) 10 kg: Scalar (it is a measure of mass, which does not include direction).

(ii) 2 meters north-west: Vector (it specifies both a magnitude, 2 meters, and a direction, north-west).

(iii) 40°: Scalar (when given alone like this, it represents only a magnitude, typically an angle, without inherent directional sense unless associated with a vector).

(iv) 40 watt: Scalar (it is a measure of power, with no directional component).

(v) 10–19 coulomb: Scalar (it is a measure of electric charge, which is not directional).

(vi) 20 m/s²: Vector (it specifies an acceleration, which is a change in velocity per unit time in a specific direction).

Question 3. Classify the following as scalar and vector quantities.

(i) time period

(ii) distance

(iii) force

(iv) velocity

(v) work done

Solution:

(i) Time Period: Scalar (it measures duration and does not involve a direction).

(ii) Distance: Scalar (it measures the length of a path between two points and is directionless).

(iii) Force: Vector (it is described by both magnitude and the direction in which it acts).

(iv) Velocity: Vector (it represents the rate of change of position and includes direction).

(v) Work Done: Scalar (it is the energy transferred when a force is applied over a distance, but it does not inherently have a directional component).

Question 4. In Fig 10.6 (a square), identify the following vectors.

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(i) Coinitial

(ii) Equal

(iii) Collinear but not equal

Solution:

(i) Coinitial: Vectors that have the same initial point or start from the same point. In diagram, coinitial vectors are and .

(ii) Equal: Vectors that have the same magnitude and direction. In diagram, equal vectors are and .

(iii) Collinear but not equal: Vectors that lie on the same line (or extended line) but differ in magnitude or direction. In diagram, collinear vectors are and .

Question 5. Answer the following as true or false.

(i) and 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.

Solution:

(i) True: and are collinear because they lie on the same line or its extension.

(ii) False: Collinear vectors must be aligned along the same line, but they can have different magnitudes.

(iii) False: Having the same magnitude does not imply that vectors are collinear.

(iv) False: This statement is not necessarily true because two collinear vectors of the same magnitude can point in opposite directions.