Vectors Introduction - Vector In Mathematics

In this post, we will talk about certain properties of vectors such as arithmetics of vector, Dot product, and Projection and will discuss how to change the basis vectors.

Basis Vector

So, you have seen some notations like 3î + 2ĵ,  The symbols î and ĵ represent basis vectors, Basis vectors are vector which defines your coordinate system.

The î is [1, 0] whereas ĵ is [0,1], They are nothing but the axis of the coordinate system where î represents X-axis and ĵ represents Y-axis.

So, 3î + 2ĵ means 3 times of î and 2 times of ĵ.

Arithmetics Of Vector

Modulus or Length of the vector.

Suppose you have a vector v = [4, 5], Then the length or modulus of the vector is represented by 

| v | = √ x² + y²

Multiplication and divide by a scaler

Vector also supports arithmetic operations such as +, - , x, / . But vectors are not normal quantities such as numbers, for example, if you do 5 x 6 you will get 30, but say you have a vector v1 and you do v1 x 5 then what will be the result?

The result will, of course, be a vector but we have to know that what happens when we multiplied 5 to a vector, does its direction change? or what about its length?

Let's look at an example,

Suppose we have v1 = [2, 2] Then if we plot it will look like,

Now let's multiply 2 to it, we get v1*2 = [ 4, 4].

So, you can observe that multiplication will scale the vector, here we multiplied v1 with 2, the resultant vector is scaled by a factor of 2.

Similarly, The divide operator will also work like that but instead of increasing, it will decrease.

Addition and Substraction

Addition in vectors are easy, suppose we have two vectors v1 = [2, 3] and v2 = [3, 2] then their addition will be neat as v1 + v2 = [ 2+3 , 3+2] = [5, 5]

same follows with subtraction.

Dot Product or inner Product

The dot product is a neat and interesting concept in vector, Suppose you have two vectors r = [ 2, 3] and s = [4, 2] Then their dot product will be 2*4 + 3*2 = 8+6 = 14.

It is represented as r . s = r1 * s1 + r2*s2 were r = [r1 , r2] and s = [s1 , s2].

Observe that the result of the dot product is a scaler value.

Orthogonal Vectors

Orthogonal vectors are the most important topic in vectors, Two vectors are said to orthogonal to each other if they make an angle of 90 degrees.

Remember the basis vector we studied above? î = [1, 0] and ĵ = [0,1], They are orthogonal to each other.

we use the formula cosϑ  = R . S / |R|*|S|

so, cosϑ = [1, 0] . [0, 1] / 1*1

cosϑ = 0
ϑ = 90 degrees.

 Vector Projection

Projection is another very fascinating topic in vectors, let's understand the projection with an example,

In the above figure, you can observe that there are two vectors S and R at an angle ϑ, the reflection of vector S in Vector R is known as the projection of S in R.

Projection of s = (R . S / R . R) * R

Change Of Basis Vector.

Look at this figure,

Here, the basis vectors are î and ĵ, We have vector Re = 3î + 4ĵ.

It is very clear that we have written Re as terms of î and ĵ i.e Re is 3 times of vector [1,0] and 4 times of vector [0,1].

Similarly, we have vectors B1 and B2 as  2î + 1ĵ and -2î + 4ĵ respectively.

Now, let's understand what do we mean by changing the basis vectors. I have written Re in terms of î and ĵ but can I write Re in terms of B1 and B2?

This is what we mean by changing the basis vectors, i.e we are shifting from î and ĵ coordinates or basis to some other basis.

Why we do that? Now, let us assume that î and ĵ are basis vector for your world i.e for the earth but what about say mars? it is obvious that their basis vectors may have some other values and in order to switch to their coordinate system, we often change our basis vectors to theirs.

We can change from one basis vectors to others in many ways and one such method is by using the concept of Projections.
But this method only works when the vectors are orthogonal to each other.

For the sake of understanding, let's take the above example, Can I write Re in terms of B1 and B2?
Let's begin the solution.

First, check if B1 and B2 are orthogonal or not.

cosϑ = [2, 1] . [-2, 4] / √10

cosϑ = 0.

S0, B1, and B2 are orthogonal i.e 90 degrees to each other.

Now, I will project Re in B1,

Projection = (Re . B1 / B1 . B1) * B1

projection = 2B1

This means that the projection of Re in B1 is 2 times of B1.

Now let's project the Re in B2.

we get projection = (1/2)B2

So, We can Re as Re = 2B1 + (1/2)B2, It is the same as 3î + 4ĵ, you can check this by expanding b1 and b2.

The concept is very neat and logical, we can just change our basis from set to others by using simple projection methods, But this works only when other vectors are orthogonal to each other.

In the real-world problem, we don't have orthogonal vectors, Then how to change the basis? we will see the next methods in further posts that will introduce you to the matrices.

Matrices are core heart for vector transformation, you will see why matrices are used, what are the purpose of matrices in further posts.

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