NCERT Class 12 Mathematics — Vector Algebra, Exercise 10.1. All 5 questions solved.

Exercise 10.1 is the definitions exercise. No calculation appears anywhere in it — every question is answered by knowing precisely what four words mean, and the whole of the rest of the chapter leans on them.

• A scalar has magnitude only. A vector has magnitude and a direction.
• Collinear vectors lie along the same line or parallel lines. They may point in opposite directions and may have different lengths.
• Coinitial vectors start from the same point, whatever they do afterwards.
• Equal vectors have the same magnitude and the same direction. Where they sit on the page is irrelevant.

Key insight. Every wrong answer in this exercise comes from collapsing two of those four ideas into one. Collinear is about the line; equal is about magnitude and direction together; coinitial is about the starting point only. Two vectors can be collinear without being equal, equal without being coinitial, and coinitial without being either.

Question 1

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

Solution. “30° east of north” means: start pointing north, then turn 30° towards the east. The angle is measured from the north direction, not from the east–west line — reading it the other way is the standard error here and gives a vector 60° out.

Choose a scale, since 40 km cannot be drawn life-size. Taking 1 cm to represent 10 km, the displacement is drawn as a 4 cm arrow from the origin O, inclined at 30° to ON:

The vector $\overrightarrow{\mathrm{OP}}$ represents the required displacement: its length carries the magnitude 40 km, and the 30° inclination to ON carries the direction.

Question 2

Classify the following measures as scalars and vectors.

Solution. The test is simply whether stating the quantity requires a direction to be meaningful.

• (i) 10 kg — mass. A mass has no direction. Scalar.
• (ii) 2 metres north-west — a displacement, and “north-west” is doing real work in the statement. Vector.
• (iii) 40° — an angle. Scalar.
• (iv) 40 watt — power. Scalar.
• (v) $10^{-19}$ coulomb — electric charge. Charge carries a sign, but a sign is not a direction in space. Scalar.
• (vi) 20 m/s² — acceleration, which is a rate of change of velocity, and velocity is directed. Vector.

Part (v) is the one that catches people: a negative charge is not a “backwards” charge in the way that $-\vec{a}$ is a backwards vector.

Question 3

Classify the following as scalar and vector quantities.

Solution.

• (i) Time period — a duration. Scalar.
• (ii) Distance — total path length travelled, with no direction attached. Scalar. Note that displacement is the vector counterpart.
• (iii) Force — has a magnitude and a line of action. Vector.
• (iv) Velocity — speed in a stated direction. Vector. (Speed alone is the scalar.)
• (v) Work done — defined as $\vec{F}\cdot\vec{d}$, a dot product of two vectors, and a dot product returns a number. Scalar.

Distance/displacement and speed/velocity are deliberately paired here: in each pair the scalar is the one that forgets the direction.

Question 4

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

Solution. Read the arrowheads first, because everything follows from them. Calling the corners P (top left), Q (top right), R (bottom right) and S (bottom left):

$$\vec{a} = \overrightarrow{\mathrm{PQ}}, \qquad \vec{b} = \overrightarrow{\mathrm{QR}}, \qquad \vec{c} = \overrightarrow{\mathrm{RS}}, \qquad \vec{d} = \overrightarrow{\mathrm{PS}}$$

(i) Coinitial. Coinitial means sharing an initial point. $\vec{a}$ begins at P and $\vec{d}$ also begins at P, so $\vec{a}$ and $\vec{d}$ are coinitial.

(ii) Equal. $\vec{b}$ runs down the right side and $\vec{d}$ runs down the left side. Both point in the same direction and, the figure being a square, both have the same length. So $\vec{b}$ and $\vec{d}$ are equal — even though they are nowhere near each other, because position is not part of what makes vectors equal.

(iii) Collinear but not equal. $\vec{a}$ points right along the top and $\vec{c}$ points left along the bottom. They are parallel, hence collinear, but opposite in direction, so $\vec{a}$ and $\vec{c}$ are collinear but not equal.

A note on part (i). The answer printed at the back of the textbook gives $\vec{a}$ and $\vec{b}$ as the coinitial pair. That does not agree with Fig 10.6 as it is drawn: $\vec{b}$ begins at Q, which is where $\vec{a}$ ends, so the two are consecutive rather than coinitial. The reading above is the one the figure supports, and it is corroborated by the textbook’s own answers to parts (ii) and (iii) — those are only correct if $\vec{b}$ and $\vec{d}$ both run downwards, which is exactly the figure that makes $\vec{a}$ and $\vec{d}$ the coinitial pair. If your school expects the printed answer, quote it; the reasoning here is what the diagram actually shows.

Question 5

Answer the following as true or false.

Solution.

(i) $\vec{a}$ and $-\vec{a}$ are collinear. True. Negating a vector reverses its direction but leaves it on the same line, and collinearity does not care about direction.

(ii) Two collinear vectors are always equal in magnitude. False. $\hat{i}$ and $5\hat{i}$ are collinear with magnitudes 1 and 5.

(iii) Two vectors having same magnitude are collinear. False. $\hat{i}$ and $\hat{j}$ both have magnitude 1 but are perpendicular.

(iv) Two collinear vectors having the same magnitude are equal. False. This is the subtle one. $\hat{i}$ and $-\hat{i}$ are collinear and both have magnitude 1, yet they are not equal, because equality needs the same direction and these are opposite. Question 4(iii) is the same point drawn as a picture.

Common mistakes

• Measuring “30° east of north” from the east axis. It is measured from north, turning towards east. Taken from the east axis the vector comes out 60° wrong, and question 1 is the only place in this exercise where a drawing can be marked wrong outright.
• Treating “collinear” as “parallel and same direction”. Collinear vectors may be antiparallel — question 5(i) and (iv) both turn on this, and so does question 4(iii).
• Assuming equal vectors must share a starting point. They need not: in question 4, $\vec{b}$ and $\vec{d}$ are equal while sitting on opposite sides of the square. Vectors are free to be translated.
• Calling charge a vector because it can be negative. A minus sign on a scalar is not a direction in space. Question 2(v).
• Confusing distance with displacement, or speed with velocity. In each pair the first is a scalar. Question 3(ii) and (iv) test exactly this.
• Reading the arrowheads carelessly in question 4. Every part of that question is decided by which way the four arrows point; identify each vector as $\overrightarrow{\mathrm{PQ}}$, $\overrightarrow{\mathrm{QR}}$ and so on before answering anything.

Practise next

• Exercise 10.2 — components, magnitude, unit vectors and the section formula, where these definitions start being used in calculations.
• Exercise 10.3 — the scalar product, and the first real use of the angle between two vectors.