What are example of vectors?

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Scalar Quantity Definition
Scalar Quantity Examples
Vector Quantity Definition
Vector Quantity Examples
Difference Between Scalar and Vector Quantity
Vector Representation
Characteristics of Vectors
Scalar and Vector Quantity Example
Subtracting
Calculations
Adding Vectors
Subtracting Vectors
Magnitude of a Vector
Vector vs Scalar
Example: kb is actually the scalar k times the vector b.
Multiplying a Vector by a Vector (Dot Product and Cross Product)
More Than 2 Dimensions
Magnitude and Direction

In mathematics and physics, we have physical quantities which can be categorized in two ways, namely

Scalar Quantity
Vector Quantity

In this article, let us discuss what are vector and scalar quantities with examples.

Scalar Quantity Definition

The physical quantities which have only magnitude are known as scalar quantities. It is fully described by a magnitude or a numerical value. Scalar quantity does not have directions. In other terms, a scalar is a measure of quantity. For example, if I say that the height of a tower is 15 meters, then the height of the tower is a scalar quantity as it needs only the magnitude of height to define itself. Let’s take another example, suppose the time taken to complete a piece of work is 3 hours, then in this case also to describe time just need the magnitude i.e. 3 hours.

Scalar Quantity Examples

Other examples of scalar quantities are mass, speed, distance, time, energy, density, volume, temperature, distance, work and so on.

Vector Quantity Definition

The physical quantities for which both magnitude and direction are defined distinctly are known as vector quantities. For example, a boy is riding a bike with a velocity of 30 km/hr in a north-east direction. Then, as we see for defining the velocity, we need two things, i.e. the magnitude of the velocity and its direction. Therefore, it represents a vector quantity.

Vector Quantity Examples

Other examples of vector quantities are displacement, acceleration, force, momentum, weight, the velocity of light, a gravitational field, current, and so on.

Difference Between Scalar and Vector Quantity

Let us discuss some difference here:

Difference Between Scalar and Vector Quantity
Scalar Quantity	Vector Quantity
It has magnitude only	It has both magnitude and the direction
It does not have direction	It has direction
It is specified by a number and a unit	It is also specified by a number along with the direction and unit
It is represented by quantity symbol	It is represented by quantity symbol in bold or with the arrow sign above
Example: Temperature, speed, etc	Example: Acceleration, velocity, etc

Vector Representation

Let us have a look at the line segment drawn below. A vector quantity always has a starting point and an endpoint. The two endpoints of the given line segment are distinguishable as and. It represents a directed line segment

The directed line segment with an initial point A and terminal point B is symbolically denoted as AB in bold.

\(\begin{array}{l}\text{Also, it can be represented as} \ \overrightarrow{AB}\end{array} \)

The length a of the vector represents its magnitude which is denoted by |AB|. Instead of using double letter notation we can use a single letter notation to represent a vector as a, b, c and it denotes their magnitudes. As it is difficult to write letters in bold we use a bar above the letters to represent vectors as ā.

Therefore,

\(\begin{array}{l}\text{If} \ \overrightarrow{AB} = a,\ \text{then} |\overrightarrow{AB}| = a, \\ \text{where} \ |\overrightarrow{AB}|\ \text{indicates the magnitude of a vector.}\end{array} \)

Also, the magnitude is called the modulus.

Characteristics of Vectors

The characteristics of the vectors are as follows:

Vectors possess magnitude as well as the direction
It does not obey the ordinary law of algebra
Either the magnitude or direction change or both change

Scalar and Vector Quantity Example

Question:

Find out the scalar and vector quantity from the given list.

Force, Speed, Electric field, Angular Momentum, Magnetic Moment, Temperature, Linear Momentum, Average Velocity.

Solution:

From the given list,

Scalar Quantities – Speed, Temperature.
Vector Quantities – Force, Electric field, Angular Momentum, Magnetic Moment, Linear Momentum, Average Velocity.

Now we are familiar with what are vectors and scalars. Now if somebody asks if acceleration is a vector or a scalar, we can easily tell that it’s a vector because it has direction as well as magnitude. Similarly, when asked if the distance is a vector or scalar, it is quite evident that as distance has only magnitude, it is a scalar quantity.

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This is a vector:

A vector has magnitude (size) and direction:

The length of the line shows its magnitude and the arrowhead points in the direction.

We can add two vectors by joining them head-to-tail:

And it doesn't matter which order we add them, we get the same result:

The two vectors (the velocity caused by the propeller, and the velocity of the wind) result in a slightly slower ground speed heading a little East of North.

If you watched the plane from the ground it would seem to be slipping sideways a little.

Have you ever seen that happen? Maybe you have seen birds struggling against a strong wind that seem to fly sideways. Vectors help explain that.

Velocity, acceleration, force and many other things are vectors.

Subtracting

We can also subtract one vector from another:

first we reverse the direction of the vector we want to subtract,
then add them as usual:

a − b

Notation

A vector is often written in bold, like a or b.

A vector can also be written as the letters
of its head and tail with an arrow above it, like this:

Calculations

Now ... how do we do the calculations?

The most common way is to first break up vectors into x and y parts, like this:

The vector a is broken up into
the two vectors ax and ay

(We see later how to do this.)

Adding Vectors

We can then add vectors by adding the x parts and adding the y parts:

The vector (8, 13) and the vector (26, 7) add up to the vector (34, 20)

c = a + b

c = (8, 13) + (26, 7) = (8+26, 13+7) = (34, 20)

When we break up a vector like that, each part is called a component:

Subtracting Vectors

To subtract, first reverse the vector we want to subtract, then add.

a = v + −k

a = (12, 2) + −(4, 5) = (12, 2) + (−4, −5) = (12−4, 2−5) = (8, −3)

Magnitude of a Vector

The magnitude of a vector is shown by two vertical bars on either side of the vector:

|a|

OR it can be written with double vertical bars (so as not to confuse it with absolute value):

||a||

We use Pythagoras' theorem to calculate it:

|a| = √( x2 + y2 )

|b| = √( 62 + 82) = √( 36+64) = √100 = 10

A vector with magnitude 1 is called a Unit Vector.

Vector vs Scalar

A scalar has magnitude (size) only.

Scalar: just a number (like 7 or −0.32) ... definitely not a vector.

A vector has magnitude and direction, and is often written in bold, so we know it is not a scalar:

so c is a vector, it has magnitude and direction
but c is just a value, like 3 or 12.4

Example: kb is actually the scalar k times the vector b.

When we multiply a vector by a scalar it is called "scaling" a vector, because we change how big or small the vector is.

a = 3m = (3×7, 3×3) = (21, 9)

It still points in the same direction, but is 3 times longer

(And now you know why numbers are called "scalars", because they "scale" the vector up or down.)

Multiplying a Vector by a Vector (Dot Product and Cross Product)

How do we multiply two vectors together? There is more than one way!

(Read those pages for more details.)

More Than 2 Dimensions

Vectors also work perfectly well in 3 or more dimensions:

The vector (1, 4, 5)

c = a + b

c = (3, 7, 4) + (2, 9, 11) = (3+2, 7+9, 4+11) = (5, 16, 15)

|w| = √( 12 + (−2)2 + 32 ) = √( 1+4+9) = √14

Here is an example with 4 dimensions (but it is hard to draw!):

(3, 3, 3, 3) + −(1, 2, 3, 4) = (3, 3, 3, 3) + (−1,−2,−3,−4) = (3−1, 3−2, 3−3, 3−4)

= (2, 1, 0, −1)

Magnitude and Direction

We may know a vector's magnitude and direction, but want its x and y lengths (or vice versa):

	<=>
Vector a in Polar Coordinates		Vector a in Cartesian Coordinates

You can read how to convert them at Polar and Cartesian Coordinates, but here is a quick summary:

From Polar Coordinates (r,θ) to Cartesian Coordinates (x,y)		From Cartesian Coordinates (x,y) to Polar Coordinates (r,θ)
x = r × cos( θ ) y = r × sin( θ )		r = √ ( x2 + y2 ) θ = tan-1 ( y / x )