# I Multiplying a vector by a complex number

1. Dec 17, 2016

### topito2

I have learned that if I multiply a vector, say 3i + 4j, by a scalar that is a real number, say 2, the effect of the operation is to expand the size of the magnitude of the original vector, by 2 in this case, and the result would be 6i + 8j.

What would be the effect on a vector, like 3i + 4j, if I multiply it by a scalar that is a complex number, say 2 + 5i?

2. Dec 17, 2016

### blue_leaf77

If you have a scalar $z$, be it real or complex, and multiply with a vector $v = \sum_{i=1}^N v_ie_i$ where $N$ the dimension of the vector, $v_i$ the i-th component which can be real or complex scalar, and $e_i$ the i-th basis (unit) vector, the result will be $zv = \sum_{i=1}^N (zv_i) e_i$.
Note that there is possible confusion in your notation, you use $i$ both for denoting the component of a vector and the imaginary number $i=\sqrt{-1}$.

Last edited: Dec 18, 2016
3. Dec 17, 2016

### Delta²

When you multiply by a complex number you essentially create a dyad of vectors $(v_1,v_2)$ one for the real part ($v_1$) and one for the imaginary part ($v_2$), where in your example it would be $v_1=2(3i+4j)=6i+8j$ and $v_2=5(3i+4j)=15i+20j$

4. Dec 18, 2016

### Svein

For this operation to have meaning, you would need to consider your vector to have complex coordinates. So in your vector, the "6" is in reality (6 + 0i) etc.You will find that most vector operations will be the same when using complex coordinates.

Except
The scalar product (dot product) will have a slightly different definition. Instead of $\vec{a}\cdot \vec{b} =\sum_{k}a_{k}b_{k}$ the definition is $\vec{a}\cdot \vec{b} =\sum_{k}a_{k}\bar{b}_{k}$ where the bar denotes complex conjugation.

5. Dec 18, 2016

### Staff: Mentor

It is probably better to rename the parts of your vector: 3x+4y. Then your multiplication gives (6+10i)x + (8+20i)y. Your vector space has to be a vector space over the complex numbers to make such a multiplication meaningful.

6. Dec 19, 2016

### topito2

thank you so much for your quick responses, guys! you have been very helpful.

7. Dec 19, 2016

### topito2

so according to what you have exposed, there is no geometrical interpretation for multiplying a vector (with real numbers as components) by a complex number? performing that kind of operation won't make the vector to have an expanded magnitude, to rotate, or something else?

8. Dec 19, 2016

### Staff: Mentor

Not that I know of.

A vector with real components is an element of a vector space over the field of real numbers. I.e., the components of the vector are real numbers. The basic operations on such a vector space are vector addition and multiplication by scalars (real numbers in this case).

A vector with complex components is an element of a vector space over the field of complex numbers. The basic operations are the same as above, except that scalar multiplication includes multiplying by complex scalars.

9. Dec 21, 2016

### topito2

Thank you so much for your help!!!!