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:

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.

(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 \vec{a} and \vec{d}.

(ii) Equal: Vectors that have the same magnitude and direction. In diagram, equal vectors are \vec{b} and \vec{d}.

(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 \vec{a} and \vec{c}.

Question 5. Answer the following as true or false.

(i) \vec{a} and -\vec{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.

Solution:

(i) True:\vec{a} and -\vec{a} 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.