Vectors+ 11 - 2D Vector · Complex scalar: UNIZOR.COM - Math+ & Problems ...

Notes to a video lecture on http://www.unizor.com

Vectors+ 11
2D Vector · Complex scalar

We know that a two-dimensional vector multiplied by a real scalar changes its length proportionally, but the direction remains the same for a positive multiplier or changes to opposite for a negative multiplier.
Let's examine how vectors on a two-dimensional plane are affected if multiplied by a non-zero complex scalar.

Our first task is to define a multiplication operation of a vector by a complex number.
To accomplish this, let's recall that any complex number a+i·b, where both a and b are real numbers and i²=−1, can be represented by a vector in a two-dimensional Euclidean space (that is, on the coordinate plane) with abscissa a and ordinate b.
This representation establishes one-to-one correspondence between vectors on a plane and complex numbers.

Using this representation, let's define an operation of multiplication of a vector on a coordinate plane {a,b} by a complex number z=x+i·y as follows:
1. Find a complex number that corresponds to our vector. So, if a vector has coordinates {a,b}, consider a complex number a+i·b.
2. Multiply this complex number by a multiplier z=x+i·y using the rules of multiplication of complex numbers. This means
(a+i·b)·z = (a+i·b)·(x+i·y) =
= a·x+a·i·y+i·b·x+i·b·i·y =
= (a·x−b·y) + i·(a·y+b·x)
3. Find the vector that corresponds to a result of multiplication of two complex number in a previous step. This vector should have abscissa a·x−b·y and ordinate a·y+b·x.
4. Declare the vector in step 3 as a result of an operation of multiplication of the original vector {a,b} by a complex multiplier z=x+i·y. So, the result of multiplication of vector {a,b} by a complex multiplier z=x+i·y is a vector {a·x−b·y,a·y+b·x}

Now we have to examine the geometric aspect of this operation.
For this, let's represent a multiplier z=x+i·y as
z = √x²+y²·(x/√x²+y² + i·y/√x²+y²)

Two numbers, x/√x²+y² and y/√x²+y², are both in the range from −1 to 1 and a sum of their squares equals to 1.
Find an angle φ such that
cos(φ)=x/√x²+y² and
sin(φ)=y/√x²+y².
Now the multiplier looks like
z = |z|·[cos(φ) + i·sin(φ)]
where |z| = √x²+y²

Using this representation, the product of a vector {a,b} by multiplier z=x+i·y looks like
{a·x−b·y,a·y+b·x} = {a',b'}
where
a' = |z|·[a·cos(φ)−b·sin(φ)] and
b' = |z|·[a·sin(φ)+b·cos(φ)]

The geometric meaning of the transformation from vector {a,b} to vector {a',b'} is a rotation of the vector by angle φ.
Here is why.
The length of a vector {a,b} is L=√a²+b².
If the angle this vector makes with an X-axis is α, the abscissa and ordinate of our vector can be expressed as
a = L·cos(α)
b = L·sin(α)

Using this representation, let's express the coordinates of a vector {a',b'} obtained as a result of multiplication of the original vector {a,b} by a complex
z=x+i·y=|z|·[cos(φ) + i·sin(φ)]
in terms of L and α.

a'=|z|·[L·cos(α)·cos(φ)−L·sin(α)·sin(φ)]
and
b'=|z|·[L·cos(α)·sin(φ)+L·sin(α)·cos(φ)]

Recall from Trigonometry:
cos(α+φ)=cos(α)·cos(φ)−sin(α)·sin(φ)
sin(α+φ)=cos(α)·sin(φ)+sin(α)·cos(φ)

Now you see that
a'=|z|·L·cos(α+φ)
b'=|z|·L·sin(α+φ)

So, the multiplication of a vector {a=L·cos(α),b=L·sin(α)} by a complex number z=x+i·y=|z|·[cos(φ)+i·sin(φ)], can be interpreted as changing the length of an original vector by a factor |z| and a rotation by angle φ.