Exercise 10.1 answers: Vector Algebra

Open NCERT p. 342One way to think about it

1. Choose a scale, say 1 cm = 10 km, so 40 km becomes a 4 cm line.
2. Draw a small compass cross at the starting point: north pointing up, east pointing right.
3. Starting from north, turn 30° towards east — that fixes the direction to draw along.
4. Draw a 4 cm line from the starting point along this direction, with an arrowhead at the far end.

In shortA 4 cm arrow (using 1 cm = 10 km) drawn 30° away from north, turning towards east, with the arrowhead showing the direction of the displacement.

Watch this explained “Forty kilometres each way”, 18:09 into The named kinds — zero, unit, coinitial, collinear, equal, negative and free

Open NCERT p. 342Matches NCERT’s answer

(i) 10 kg

1. Mass is 'how much matter', it does not point anywhere.
2. So 10 kg has no direction — it is a scalar.

AnswerScalar

(ii) 2 meters north-west

1. This gives both a size (2 m) and a direction (north-west).
2. A quantity with size and direction is a vector.

AnswerVector

(iii) 40°

1. 40° is a plain number of degrees, an angle, with no direction of its own.
2. So it is a scalar.

AnswerScalar

(iv) 40 watt

1. Watt is the unit of power, and power has no direction.
2. So 40 watt is a scalar.

AnswerScalar

(v) 10⁻¹⁹ coulomb

1. Coulomb is the unit of electric charge, and charge has no direction.
2. So it is a scalar.

AnswerScalar

(vi) 20 m/s²

1. m/s² is the unit of acceleration, and acceleration always points somewhere.
2. So it is a vector.

AnswerVector

Watch this explained “The shortcut that fails twice”, 12:07 into Why some measurements need a direction attached and others do not

Open NCERT p. 342Matches NCERT’s answer

(i) time period

1. Time period is just a duration in seconds, with no direction.
2. So it is a scalar.

AnswerScalar

(ii) distance

1. Distance is the total length of the path covered — it does not point anywhere.
2. So it is a scalar.

AnswerScalar

(iii) force

1. A force always pushes or pulls in some direction.
2. So force is a vector.

AnswerVector

(iv) velocity

1. Velocity is speed together with a direction of motion.
2. So it is a vector.

AnswerVector

(v) work done

1. Work done comes out as a single number (force times distance moved), with no direction of its own.
2. So it is a scalar.

AnswerScalar

Watch this explained “The two lists”, 1:00 into Why some measurements need a direction attached and others do not

Open NCERT p. 342Checked by computer

(i) Coinitial

1. Coinitial vectors start from the same point.
2. a⃗ (top side) and d⃗ (left side) both start at the top-left corner.

Answera⃗ and d⃗

(ii) Equal

1. Equal vectors must have the same length and point the same way.
2. Every side of a square has the same length.
3. b⃗ (right side) and d⃗ (left side) both point downward, so they match in length and direction.

Answerb⃗ and d⃗

(iii) Collinear but not equal

1. Collinear vectors lie along parallel lines, whichever way they point.
2. a⃗ (top, pointing right) and c⃗ (bottom, pointing left) lie along parallel lines but point opposite ways.
3. So they are collinear, but not equal.
4. The answer key at the back of the book says a⃗ and b⃗ are coinitial, but b⃗ starts at the corner where a⃗ ends; the two arrows that start from the same corner (top-left) are a⃗ and d⃗.

Answera⃗ and c⃗

Watch this explained “The same three questions, on a square”, 14:36 into The named kinds — zero, unit, coinitial, collinear, equal, negative and free

Open NCERT p. 342Matches NCERT’s answer

(i) are collinear.

1. Collinear vectors lie along the same or a parallel line.
2. a⃗ and −a⃗ point in exactly opposite directions along the same line.
3. So they are collinear.

AnswerTrue

(ii) Two collinear vectors are always equal in magnitude.

1. Collinear only means 'along parallel lines' — it says nothing about length.
2. Example: (1,0,0) and (2,0,0) are collinear but have different lengths.

AnswerFalse

(iii) Two vectors having same magnitude are collinear.

1. Same magnitude only means same length — it says nothing about direction.
2. Example: (1,0,0) and (0,1,0) have equal length but do not lie along the same line.

AnswerFalse

(iv) Two collinear vectors having the same magnitude are equal.

1. Collinear vectors of the same length can still point opposite ways.
2. Example: (1,0,0) and (−1,0,0) are collinear with equal length but point opposite ways, so they are not equal.

AnswerFalse

Watch this explained “Four claims to judge”, 15:51 into The named kinds — zero, unit, coinitial, collinear, equal, negative and free