Vectors and Scalars

Physical quantities which can be completely described in terms of a single numerical value with a unit are called scalar quantities.

Physical quantities associated with both numerical value and direction and can not be described by a single number are called vector quantities

For example, you want to measure the temperature of your body with a thermometer, you never consider the direction of the temperature but only consider the single numerical value in 'degrees F' or 'degrees C'. So, temperature is a scalar quantity.

The physical quantities such as time, speed, mass, volume, density, pressure, energy etc. are scalar quantities. Some scalars are always positive such as speed, mass while others can be positive or negative such as temperature.

Adding or subtracting scalar quantities is the same thing as adding or subtracting numbers.

A physical quantity which can be completely described by a single number with a unit is called scalar.

In case of a flying airplane, you not just consider how fast the plane is moving but you must also consider in which direction the plane is moving. The velocity is a physical quantity which has both numerical value and direction which can completely describe the motion of a plane and velocity is a vector quantity.

The physical quantities such as displacement, velocity, force etc. are vector quantities.

A physical quantity which can be completely described by a number with a unit (magnitude) and direction is called vector.

Adding or subtracting vectors is not the same as adding or subtracting scalar values. The direction of vectors must be considered in vector addition or subtraction

If a physical quantity is associated with direction, we call that quantity a vector quantity and if it is not associated with any direction, we call it a scalar quantity.

A direction is always associated with velocity so it is a vector quantity but speed is not a vector quantity as there is no direction associated with it, so speed is a scalar quantity. Note that a vector quantity has both magnitude and direction but a scalar quantity has magnitude only.

How do we represent a vector? Let a vector quantity be represented by a letter say, J and to write this as a vector we put an arrow over it like, $\vec{J}$ where the arrow represents the direction and the letter alone represents the magnitude.

If you are writing the magnitude of a vector quantity alone, you don't need to put the arrow over it but to write it as a vector with both magnitude and direction you need to put the arrow over the symbol of the physical quantity.

Now we consider vectors in a more general way. The graphical way of representing a vector is an arrow that starts at a particular point (say $A$) and ends at another point (say $B$) as in Figure 1.

The vector in Figure 1 can be written as $\vec{AB}$ (read as "vector $AB$") in notation but the vector $\vec{AB}$ can also be represented by a single letter like $\vec{A}$ (Figure 2). The way to write a vector by a single letter with an arrow over it is more preferred way.

Figure 1 A vector represented graphically by an arrow beginning at a point $A$ (tail) and ending at a point $B$ (head).
Figure 2 The vector $\vec{AB}$ in Figure 1 is written as $\vec{A}$ in a more general way.

Now you should know that the physical quantities having both magnitude and direction are vectors while those having only the magnitude but not the direction are scalars.

The scalars are added or subtracted in the same way as adding or subtracting common numbers but this is not the case for vectors because we also need to consider the direction. The mass is a scalar quantity and you can directly add $5\text{kg}$ and $4\text{kg}$ to get the total mass $9\text{kg}$ but this is not applicable for vectors.

Unit Vectors

The work of unit vectors is to point a particular direction in space. Unit vectors are written by using the "hat" or "cap" sign over the physical quantity such as $\hat{n}$ read as "n cap".

See in Figure 3 that the unit vectors that point in the positive x-direction, positive y-direction and positive z-direction are represented by $\hat{i}$, $\hat{j}$ and $\hat{k}$ respectively.

Figure 3 The unit vectors shown along positive x-axis, positive y-axis and positive z-axis.

The "cap" or "hat" sign is the reminder of the unit vectors. Mathematically the unit vectors can be obtained by dividing a vector by its magnitude. For example consider a unit vector say $\hat{a}$ of a vector $\vec{A}$ as shown in Figure 2. The unit vector $\hat a$ is $\vec{A}$ divided by its magnitude $\left| \vec{A} \right|$ or $A$. Therefore,

\[\hat{a}=\frac{\vec{A}}{\left| \vec{A} \right|}=\frac{\vec{A}}{A}\]

Tapes of Vectors

The unit vector is also one of the types of vectors which is already noted. The equal vectors are those which have the same direction and the same magnitude.

Figure 4 The equal vectors will have the same magnitude and direction.
Figure 5 The vectors having the same magnitude but not the same direction are not equal. The vectors shown are anti-parallel vectors, it means one vector is the negative of the another.

The Figure 4 shows equal vectors where $\vec{A}$ is equal to $\vec{B}$. Are the vectors in Figure 5 are equal if the magnitudes of both vectors are equal? Note that the vectors will be equal if the direction and the magnitude of both vectors are the same.

In Figure 5 the vectors have opposite direction so they are not equal vectors even if they have the same magnitude. The vectors which are parallel are parallel and anti-parallel are anti-parallel vectors.

Figure 6 The vectors are parallel but do not have the same magnitude. $\vec{B}$ is equal to $\vec{A}$ only when $\vec{A}$ is multiplied by a certain scalar value such as $k$ which makes the magnitude of $\vec{A}$ equal to the magnitude of $\vec{B}$.
Figure 7 The vectors are anti-parallel and do not have the same magnitude. Here one of the vectors such as $\vec{A}$ is multiplied by a negative scalar, $-k$ to make the magnitude and direction of $\vec{A}$ to be equal to that of $\vec{B}$.

The vectors in Figure 4 are parallel(also equal) and those in Figure 5 are anti-parallel vectors. Equal vectors are always parallel. Notice in Figure 6 that the vectors are only parallel(but not equal), that is they have the same direction but different magnitudes.

In this case one vector is equal to the scalar multiplication of the other. In other words, the magnitude of one vector can be brought equal to the magnitude of the other by multiplying one of the vectors by a scalar value say $k$. Therefore, we can also write $\vec{B}=k\vec{A}$ for parallel vectors as in Figure 6 and Figure 4. In the case of equal vectors the scalar has the value of $1$.

In Figure 7 the vectors are anti-parallel which have opposite direction and different magnitudes. In this case one vector can be brought equal to the other by multiplying one of the vectors by a negative scalar like $-k$.

The multiplication by $k$ fixes the magnitude and the negative sign fixes the direction. So for the anti-parallel vectors we can write $\vec{B}=-k\vec{A}$. In the case of anti-parallel vectors as in Figure 5, the scalar has the value of $1$, that is one vector is equal to the negative of another.

Note that the negative of a vector has equal magnitude but opposite direction, for example $-\vec{A}$ has the same magnitude of $\vec{A}$ but has opposite direction.

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