An auto-rickshaw leaves a hotel in Jaipur. It drives \(3\,\text{km}\) east along one road, turns left, and drives \(4\,\text{km}\) north to the railway station. The meter charges for \(7\,\text{km}\). On the map, the station is only \(5\,\text{km}\) from the hotel in a straight line. Both numbers are correct. They answer two different questions.
Numbers that are complete
Some quantities are fully described by a number and a unit. A mass of \(2\,\text{kg}\), a time of \(5\,\text{s}\), a temperature of 30 °C and an energy of \(10\,\text{J}\) need nothing more. Such a quantity is a scalar. Two scalars of the same kind add as ordinary numbers. A \(2\,\text{kg}\) bag and a \(3\,\text{kg}\) bag together always have a mass of \(5\,\text{kg}\).
Quantities that need a direction
Other quantities are not complete without a direction. A displacement of \(4\,\text{km}\) could end at the station or at the airport. A force of \(50\,\text{N}\) could lift a bag or push it down. Such a quantity is a vector. Its size is called its magnitude, and the magnitude is never negative. A vector is drawn as an arrow. The length of the arrow shows the magnitude on a chosen scale, and the arrow points in the direction of the quantity.
A scalar is a physical quantity that has magnitude only; scalars of one kind add as ordinary numbers. A vector is a physical quantity that has both magnitude and direction, and that adds by the triangle rule of the next lesson.
In print a vector is written in bold, \(\mathbf{A}\). By hand it is written with an arrow, \(\vec A\). The magnitude is written \(|\vec A|\), or simply \(A\) in italics without the arrow. An arrow of length one that points along \(\vec A\) is called the unit vector along \(\vec A\), written \(\hat A\) and read "A hat". A unit vector carries a direction and nothing else, so it has no unit. Every vector is its magnitude times its unit vector.
Distance and displacement
The meter of the auto-rickshaw measures the length of the road it covers. That length is the distance, a scalar. The \(3\,\text{km}\) and the \(4\,\text{km}\) simply add to \(7\,\text{km}\). The map measures the straight arrow from the hotel to the station. That arrow is the displacement, a vector. Its two parts are at right angles, so its magnitude is \(\sqrt{3^2 + 4^2} = 5\,\text{km}\).
The distance of a journey can only grow. The displacement can grow, shrink and turn, because it depends only on where the body is now and where it started. At the far corner of the track the displacement is \(\sqrt{80^2 + 40^2} = 89\,\text{m}\). Back at the start it is zero, although the walker has covered \(240\,\text{m}\).
A direction alone does not make a quantity a vector. An electric current flows along a wire, yet two currents that meet at a junction simply add as numbers, whatever the angle between the wires. So current is a scalar. A large rotation is not a vector either. Turn a book through \(90°\) about two different axes in two different orders, and it ends in two different positions. A vector must add by the triangle rule, and in any order.
Which of these quantities is a vector?
A current of \(2\,\text{A}\) in one wire and a current of \(3\,\text{A}\) in another wire flow into a junction at an angle of \(60°\). How much current leaves the junction?
Equal vectors
Two vectors are equal when they have the same magnitude and the same direction. Their position does not matter. Two Metro trains running side by side on parallel tracks at \(20\,\text{m/s}\) have equal velocities, although they are in different places. So a vector may be slid anywhere in a drawing, as long as its length and direction stay the same. The next lesson uses this freedom to add vectors.
The negative of a vector and the zero vector
The vector \(-\vec A\) has the same magnitude as \(\vec A\) and points the opposite way. A velocity of \(20\,\text{m/s}\) east has the negative \(20\,\text{m/s}\) west. A vector added to its own negative gives a vector of magnitude zero. This is the zero vector, written \(\vec 0\). The walk round the park ends with a zero displacement.
Two vectors that point the same way are called parallel. Two that point opposite ways are antiparallel. The angle between two vectors is measured with their tails together, and it lies between \(0°\) and \(180°\). Parallel vectors are at \(0°\), antiparallel vectors at \(180°\), and perpendicular vectors at \(90°\).
Two cars on the Mumbai-Pune expressway move at \(25\,\text{m/s}\), one in each carriageway, in opposite directions. Which statement is correct?
A vector times a number
A vector can be multiplied by an ordinary number \(k\). The product \(k\vec A\) has a magnitude \(|k|\) times that of \(\vec A\). If \(k\) is positive, \(k\vec A\) points along \(\vec A\). If \(k\) is negative, it points the opposite way. At \(k = 0\) it is the zero vector. The number \(k\) may carry a unit of its own. Then the product is a new physical quantity.
Physics is full of such products. Momentum is \(\vec p = m\vec v\), and the mass \(m\) is positive, so momentum always points along the velocity. The weight of a body is \(m\vec g\), along the downward \(\vec g\). The force on a charge \(q\) in an electric field is \(\vec F = q\vec E\). For an electron, \(q\) is negative, so the force points against the field.
The average velocity is the plainest example. It is a displacement divided by a time. The derivation below shows why it must point along the displacement.
\[ \vec v_{avg} = \dfrac{\Delta\vec r}{\Delta t} \]
Why. the average velocity is the displacement \(\Delta\vec r\) over the time interval \(\Delta t\).
\[ \vec v_{avg} = k\,\Delta\vec r, \quad k = \dfrac{1}{\Delta t} \]
Why. dividing by \(\Delta t\) is the same as multiplying by the number \(1/\Delta t\).
\[ k > 0 \]
Why. a time interval is always positive, so \(k\) is a positive number, with the unit 1/s.
\[ \vec v_{avg} \parallel \Delta\vec r, \quad |\vec v_{avg}| = \dfrac{|\Delta\vec r|}{\Delta t} \]
Read it. a positive number keeps the direction and scales the magnitude. The average velocity points along the displacement.
A vector \(\vec A\) has magnitude \(4\,\text{N}\) and points north. What is \(-3\vec A\)?
An electron, of charge \(-1.6\times10^{-19}\,\text{C}\), is in an electric field of \(1000\,\text{N/C}\) that points east. What is the force \(\vec F = q\vec E\) on it?
This is the worked example "Out, then partly back" of Motion in One Dimension, lesson 3, done with the tool. A particle starts at \(x = 0\). It moves to \(x = +5\,\text{m}\) in \(2\,\text{s}\). It then turns back and reaches \(x = +2\,\text{m}\) after another \(3\,\text{s}\). Find the average speed and the average velocity for the whole \(5\,\text{s}\).
Distance is a scalar, so the lengths of the two legs add as numbers. Displacement is a vector, here along one line, so it carries a sign. Take the positive direction to the right.
Distance \(= 5 + 3 = 8\,\text{m}\). Displacement \(= x_{final} - x_{initial} = 2 - 0 = +2\,\text{m}\).
Average speed \(= 8/5 = 1.6\,\text{m/s}\). Average velocity \(= (+2\,\text{m})/(5\,\text{s}) = +0.4\,\text{m/s}\).
The average speed is \(1.6\,\text{m/s}\) and the average velocity is \(+0.4\,\text{m/s}\), to the right, the same answers as the physics chapter. The average velocity points along the displacement, as the derivation above requires.
An auto-rickshaw drives \(6\,\text{km}\) east and then \(8\,\text{km}\) north. Find the magnitude of its displacement, in km.
A walker goes once round a track and ends where she started. Why is her distance \(240\,\text{m}\) while her displacement is zero?
Distance is a scalar. It adds the length of every stretch of the path, so it grows with every step and reaches \(240\,\text{m}\). Displacement is a vector from the starting point to the present position. It depends only on those two points, not on the path between them. When the walker returns to the start, the two points coincide, so the displacement is the zero vector.
The auto-rickshaw of the opening can now be read. The meter adds two scalars, \(3 + 4 = 7\,\text{km}\), because it measures the road. The map shows one vector, the displacement from the hotel to the station. Its magnitude is \(5\,\text{km}\), and it points \(53°\) north of east, since \(\tan^{-1}(4/3) = 53°\).
Motion in One Dimension begins with this distinction: distance and speed are scalars, while displacement and velocity are vectors. Motion in Two Dimensions writes position and displacement as arrows in a plane. In Gravitation and in Electrostatics, the potentials of several bodies add as plain numbers. Their fields are vectors and add as arrows. Electric Flux and Gauss's Law turns an area into a vector, with its direction along the normal to the surface.
- Calling a quantity a vector because it has a direction, like the current in a wire.A vector must also add by the triangle rule. Currents at a junction add as numbers, so current is a scalar.
- Writing a negative magnitude, such as \(|\vec v| = -5\,\text{m/s}\).A magnitude is never negative. A minus sign belongs to a component or to a direction, never to the size.
- Treating two vectors as unequal because they are drawn in different places.Only the magnitude and the direction decide equality. An arrow may be slid parallel to itself.