Table of Contents

Scalars

  • Some quantities in physics, such as time, distance, mass, speed and temperature, just need one number to specify them. These are called scalar quantities.

  • For example, it is sufficient to say that the mass of a body is 52 kg or that the temperature is 25.0 $^{\circ}$C.

Definition

Scalars are quantities that have magnitude only.


Table: Some scalar quantities (with SI unit).

Scalar Quantity SI Unit Symbol
Time s $t$
Mass kg $m$
Density kg/m$^3$ $\rho$
Distance m $d$
Speed m/s $s$
Energy J $E$
Power J/s $P$
Electric Charge C $q$
Current A $I$

  • Scalars may be added by simple arithmetic.

EXAMPLE : Identify the scalar quantities.

Which list contains only scalar quantities?

A  mass, acceleration, temperature, kinetic energy
B  mass, volume, electric potential, kinetic energy
C  acceleration, temperature, volume, electric charge
D  moment, electric field, density, magnetic flux

Solution (click to expand)
B

Vectors

  • Some quantities are fully specified only if, in addition to a number, a direction is also given.

  • For example, saying that you are sitting in a car travelling at 120 km/h does not tell us where you will be in 10 minutes because we do not know the direction in which you are travelling. Therefore, we need to specify the velocity, instead of just the speed.

Definition

Vectors are quantities that have direction as well as magnitude.


Table: Some vector quantities (with SI unit).

Vector Quantity SI Unit Symbol
Displacement m $\bm{s}$
Velocity m/s $\bm{v}$
Acceleration m/s$^2$ $\bm{a}$
Force N $\bm{F}$
Momentum Ns $\bm{p}$
Moment Nm $\bm{M}$
Electric field V/m $\bm{E}$

  • Vectors are denoted by boldface letters (e.g. $\bm{v}$, $\bm{a}$, $\bm{F}~$), or with an arrow cap (e.g. $\vec{v}$, $\vec{a}$, $\vec{F}~$) or with a tilde underneath (e.g. $\utilde{v}$, $\utilde{a}$, $\utilde{F}~$).

  • A vector is represented by a straight arrow as shown in the following figure. The direction of the arrow represents the direction of the vector and the length of the arrow represents the magnitude of the vector.

Figure - Vectors are represented by arrows.

  • The vectors in the following figure are all equal to each other. We therefore deduce that vectors do not have to start from the same point to be equal.

Figure - Equal vectors.

Tip

All vectors with the same magnitude and direction are equal.


  • Every vector has an initial point and a final point where the direction is from the former to the latter. Let the initial point be denoted by $A$ and the final point be denoted by $B$, then an alternative symbol for the vector is $\overrightarrow{AB}$.

Figure - A vector from point $A$ to $B$ can be represented as $\overrightarrow{AB}$.


Magnitude of a Vector

  • The magnituide of a vector $\bm{a}$ is given as either $a$ or $|\bm{a}|$.

Multiplication of a Vector by a Scalar

  • A vector can be multiplied by a number. The vector $\bm{a}$ multiplied by the scalar $2$ gives a vector in the same direction as $\bm{a}$ but $2$ times longer.

  • The vector $-\bm{a}$ has the same magnitude as $\bm{a}$ but is opposite in direction.The vector $\bm{a}$ multiplied by $−\frac{1}{2}$ is opposite to $\bm{a}$ in direction and half as long.

Figure - Multiplication of a vector by a scalar.

Addition of Vectors

  • In this article, we deal only with coplanar vectors which are vectors that lie in the same plane.

  • Given two vectors $\bm{a}$ and $\bm{b}$, we can add them in two ways - parallelogram law or triangle law for vector addition.

  • Parallelogram law for vector addition

    • we can shift the two vectors so that the initial points coincide at $O$.
    • the vectors $\bm{a}$ and $\bm{b}$ then form the sides of a parallelogram.
    • the diagonal of the parallelogram is then the resultant vector $\bm{a} + \bm{b}$.

  • Triangle law for vector addition

    • place the initial point of the second vector (i.e. $\bm{b}$) at the final point of the first vector (i.e. $\bm{a}$).
    • construct the vector pointing from the initial point of $\bm{a}$ to the final point of $\bm{b}$. This vector is the resultant vector $\bm{a} + \bm{b}$.

(a) Two vectors $\bm{a}$ and $\bm{b}$.
(b) Parallelogram law for vector addition.
(c) Triangle law for vector addition.

Figure - Constructing $\bm{a} + \bm{b}$ using both (a) parallelogram law and (b) triangle law.

  • Conversely, for the triangle law, we can also place the initial point of $\bm{a}$ at the final point of $\bm{b}$. The resultant vector $\bm{b} + \bm{a}$ constructed will be the vector pointing from the initial point of $\bm{b}$ to the final point of $\bm{a}$, which is identical to $\bm{a} + \bm{b}$.

  • this tells us that vector addition is commutative:

$$\boxed{\bm{a} + \bm{b} = \bm{b} + \bm{a} }$$

Figure - Vector addition is commutative: $\bm{a} + \bm{b} = \bm{b} + \bm{a} $ .

Subtraction of Vectors

  • Given two vectors $\bm{a}$ and $\bm{b}$, we wish to compute $\bm{a} - \bm{b}$. We realize that $~\bm{a} - \bm{b} = \bm{a} + (-\bm{b})$.

  • The above result tells us we can add $-\bm{b}$ to $\bm{a}$ using the triangle law. See Figure (b) below.

  • Further, according to our commutative law for vector addition, $\bm{a} + (-\bm{b}) = -\bm{b} + \bm{a}$. This means we can also add $\bm{a}$ to $-\bm{b}$ using the triangle law. See Figure (c) below.

(a) Two vectors $\bm{a}$ and $\bm{b}$.
(b) $~\bm{a} - \bm{b} = \bm{a} + (-\bm{b})$
(c) $~\bm{a} + (-\bm{b}) = -\bm{b} + \bm{a}$.

Figure - Two ways to construct $\bm{a} - \bm{b}$.