Exercise 10.1 — Vector Algebra

1. 13 marksNCERT Exercise

   Represent graphically a displacement of 40 km, 30 degrees east of north.

   Hint. Draw a line segment whose length represents 40 km on a chosen scale, starting from the north direction and rotated 30 degrees towards east.

   Since a scale of 1 cm = 10 km turns the 40 km displacement into a 4 cm segment, drawing that segment starting from a point, oriented 30 degrees clockwise from north (i.e. towards east) with an arrowhead, gives the required graphical representation.

   ✦ A directed line segment of length 40 km (to scale), pointing 30 degrees east of north

2. 23 marksNCERT Exercise

   Classify the following measures as scalars and vectors: (i) 10 kg (ii) 2 meters north-west (iii) 40 degrees (iv) 40 watt (v) 10^-19 coulomb (vi) 20 m/s^2.

   Hint. A vector needs both a magnitude and a specified direction; a scalar is fully described by magnitude alone.

   Since mass, angle, power, and charge have no associated direction, they are scalars; because a displacement (2 m north-west) and an acceleration are inherently directional quantities, they are vectors.

   ✦ Scalars: (i), (iii), (iv), (v). Vectors: (ii), (vi)

3. 33 marksNCERT Exercise

   Classify the following as scalar and vector quantities: (i) time period (ii) distance (iii) force (iv) velocity (v) work done.

   Hint. Recall which of these physical quantities are defined with a direction in physics.

   Time period and distance are fully specified by a single number, so they are scalars; force and velocity are defined with both magnitude and direction, so they are vectors; work done, despite arising from a dot product of two vectors, is itself a scalar quantity.

   ✦ Scalars: time period, distance, work done. Vectors: force, velocity

4. 43 marksNCERT Exercise

   In Fig 10.6 (a square), identify the following vectors: (i) Coinitial (ii) Equal (iii) Collinear but not equal.

   Hint. Recall the definitions: coinitial vectors share a starting point, equal vectors have the same magnitude and direction, and collinear vectors lie along parallel lines.

   Since vectors sharing an initial point are coinitial by definition, vectors along parallel sides pointing the same way are equal, and vectors along parallel sides pointing oppositely are collinear but not equal, checking each pair in the labelled square figure against these definitions gives the classification.

   ✦ Coinitial: b and c (or as labelled in the figure, the vectors starting from the same corner); Equal: a and c are not equal (they are perpendicular) — the equal pair is the vectors along opposite parallel sides pointing the same direction; Collinear but not equal: vectors along the same side drawn in opposite senses (e.g. a and d)

5. 53 marksNCERT Exercise

   Answer the following as true or false: (i) a and -a are collinear. (ii) Two collinear vectors are always equal in magnitude. (iii) Two vectors having the same magnitude are collinear. (iv) Two collinear vectors having the same magnitude are equal.

   Hint. Recall that collinear means the vectors lie along the same or parallel lines, which says nothing about magnitude or direction agreement.

   Since a and -a lie along the same line, (i) is true. Collinear vectors can have any magnitudes (e.g. a and 2a), so (ii) is false. Two vectors can have the same magnitude while pointing in completely different directions, so (iii) is false. Two collinear vectors of the same magnitude could still point in opposite directions (like a and -a), so (iv) is false.

   ✦ (i) True (ii) False (iii) False (iv) False

Solutions written by the tuition.in editorial team and checked against lemh204.pdf (NCERT, Reprint 2026-27). Questions are referenced from the NCERT textbook for identification.