Can two cars have the same speed but different velocities?

Why Chapter 3 needs this at all

In Chapter 2 everything moved along one line, and a plus or minus sign handled direction. In a plane, two signs can't describe "north-east" or "30° above the horizontal". You need a quantity that carries direction properly. That is a vector, and Section 3.2 sets the rules.

Scalars: one number and a unit

A scalar is fully described by a single number with its proper unit. Mass, time, temperature and the distance between two points are all scalars. NCERT's examples are small and concrete. A rectangle with sides 1.0 m and 0.5 m has a perimeter of 1.0 + 0.5 + 1.0 + 0.5 = 3.0 m. If a day's maximum and minimum temperatures are 35.6 °C and 24.2 °C, the difference is 11.4 °C.

Scalars follow ordinary arithmetic, with one restriction from the book's footnote. You can add or subtract only quantities with the same unit. Multiplying or dividing different units is fine. An aluminium cube of side 10 cm has a volume of 10⁻³ m³ and a mass of 2.7 kg. Divide, and its density is 2.7 × 10³ kg m⁻³.

Vectors: the full definition

NCERT asks three things of a vector. It has a magnitude. It has a direction. And it obeys the triangle law of addition (or the parallelogram law, which is equivalent). Displacement, velocity, acceleration and force all qualify.

Why insist on the addition law? Because the real test of a vector is how it combines with others. The next lesson covers that law properly.

In print a vector is set in bold, v. In your notebook, put an arrow over the letter. The magnitude is written |v| or simply v.

Displacement doesn't care about the path

Pick an origin O. The position vector r of a point P is the arrow from O to P. If the object moves from P to P′, its displacement is the arrow from P to P′.

Fig. 3.1(b) makes the key point. Three different routes from P to Q (PABCQ, PDQ and PBEFQ) all have the same displacement: the single straight arrow PQ. The path length changes from route to route. The displacement doesn't.

Write the rule exactly as NCERT does: the magnitude of displacement is less than or equal to the path length. Equal when the motion is along a straight line without turning back. Less otherwise. Writing "always less than" is wrong, because the straight-line case breaks it.

When are two vectors equal?

Two vectors are equal only if they have the same magnitude and the same direction. To test it, slide one parallel to itself until the tails meet. If the tips meet too, they are equal.

Sliding is allowed because these are free vectors: where they sit doesn't change them. A footnote in the book points out that some situations need localised vectors, where the location or line of application matters. Think of a force on a door: a push near the hinge and the same push near the handle do very different things.

Multiplying by a number

Multiply A by a positive number λ and you get a vector λA in the same direction, λ times as long. Multiply by a negative number and the direction flips. Fig. 3.3 draws 2A, −A and −1.5A side by side, and −1.5A is the one to look at: one and a half times as long, pointing the other way.

The number λ can carry its own unit. Velocity in m s⁻¹ times a time in seconds gives displacement in metres. The direction stays the velocity's direction.

Check yourself

A cyclist's velocity is 5 m s⁻¹ due north. What is −2 times that velocity? (Answer at the bottom.)

Questions students ask

Answer: 10 m s⁻¹ due south. The 2 doubles the size; the minus sign turns it round.