Class 12 NCERT Solutions- Mathematics Part ii – Chapter 10 – Vector Algebra Exercise 10.1
Last Updated : 01 May, 2024
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:
- Choose a Scale: For example, 1 cm = 10 km. Hence, 40 km = 4 cm.
- Draw the Axes: Align the north direction with the positive y-axis and the east direction with the positive x-axis.
- Determine the Angle: The displacement is 30° east of north.
- Plot the Displacement: From the origin, use a protractor to measure 30° from the north towards the east.
- Measure the Distance: Measure 4 cm along the 30° line from the origin.
- 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.
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