Solving Problems Involving Complex Vectors

In summary, for problem 1, the vector moduli were calculated using the formula |Vector modulus| = √(a^2 + b^2), where a is the real part and b is the imaginary part of the vector. The vector moduli for a, b, c, and d were found to be 3.605551275463989, 7.211102550927978, 7.211102550927978, and 5, respectively. For problem 2, the system of vectors was given as 3a-2b=7 (Eq1) and -5a+6b=3i (Eq2). To find the vectors a and b, Eq(1)
  • #1
Martin Harris
103
6
Homework Statement
1) The following vectors are given written as complex numbers:
a = 3-2i
b= -6-4i
c= 4+ 6i
d= -4+3i
The requirements: To calculate the vectors' modulus.

2) The following system of vectors is given
3a-2b=7
-5a+6b=3i
The requirements: To find a and b as vectors, as well as to calculate their modulus.
Relevant Equations
Let the vectors be of the following type: z = a+bi, where a = real part, b = imaginary part
Hi

Here is my attempt at a solution for problems 1) and 2) that can be found within the summary.

Problem 1)
a = 3-2i
b= -6-4i
c= 4+ 6i
d= -4+3i

Now, to calculate each vector modulus, I applied the following formula:

$$\left| Vector modulus \right| = \sqrt {(a^2 + b^2) }$$
where a = real part of the vector, and b = imaginary part of the vector

By this formula, the following vector modulus, were calculated:
$$a = \sqrt {13} = 3.605551275463989$$
$$b = 2* \sqrt {13} = 7.211102550927978$$
$$c = 2*\sqrt {13} = 7.211102550927978$$
$$d = \sqrt {25} = 5$$

End of Problem 1 solution attempt

Problem 2)
The following system of vectors is given
$$3a-2b=7 (Eq1)$$
$$-5a+6b=3i (Eq2)$$

I am requested to find vectors a,b as well as to calculate their modulus

From Eq(1), $$ vector a = \frac {7a+2b} {3} Eq (3) $$
Substituting vector a with the above form in Eq 2 yields:
$$vector b = 4.375+1.125i$$ or $$vector b = \frac {35} {8} + \frac {9i} {8} $$
Now plugging back vector b in Eq (3) yields:
$$vector a = 5.25 +0.75i$$ or $$vector a = \frac {15.75} {3} + \frac {2.25i} {3} $$

Now, to calculate the vector modulus, the same formula that was used in Problem1 will be applied, hence it will yield the following vector modulus:

$$Modulus for vector a = \frac {15*\sqrt {2}} {4} = 5.303300858899107 $$
$$Modulus for vector b = 4.517327749898163 $$

End of Problem 2 solution attempt

I would be more than grateful if someone could peer-review (check my attempt).
Many thanks!
 
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  • #2
It seems all right but would be more accurate to write it using root e.g.
[tex]a=\sqrt{13}[/tex]
 
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  • #3
Cool! Thanks for the advice, I'll follow it.
 
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Related to Solving Problems Involving Complex Vectors

1. What is a complex vector?

A complex vector is a mathematical object that consists of a set of complex numbers arranged in a specific order. It can be represented as a column or row matrix and is often used to represent quantities that have both magnitude and direction, such as forces or electric fields.

2. How do you manipulate complex vectors?

Complex vectors can be manipulated using basic vector operations such as addition, subtraction, and scalar multiplication. These operations are performed on the individual components of the vector, which are complex numbers. Additionally, complex vectors can also be multiplied using the dot product or cross product, depending on the context.

3. What is the significance of complex vectors in science?

Complex vectors are widely used in science, particularly in fields such as physics, engineering, and mathematics. They are used to represent physical quantities that have both magnitude and direction, making them useful for solving problems involving forces, electric and magnetic fields, and other complex systems.

4. Can complex vectors be visualized?

Yes, complex vectors can be visualized in a two-dimensional or three-dimensional coordinate system, similar to how we visualize regular vectors. In two dimensions, they can be represented as arrows with a specific length and direction. In three dimensions, they can be represented as arrows in space with a specific magnitude and direction.

5. Are there any real-world applications of manipulating complex vectors?

Yes, there are many real-world applications of manipulating complex vectors. Some examples include analyzing the behavior of electric and magnetic fields, designing electrical circuits, and solving problems in fluid mechanics. They are also used in computer graphics and animation to represent and manipulate 3D objects.

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