# Complex Geometry - Vectors and Paralellogram

• kathrynag
In summary, the area of a parallelogram made by vectors a and b, taken with a certain sign, is I(a conjugate)b. The sign depends on the difference between the angles of the vectors, and can be represented in the complex plane by converting the vectors into polar form. It is important to use the absolute value when considering the sign of the cross product, as it depends on the order of the vectors rather than any property of the parallelogram.
kathrynag

## Homework Statement

Lay down vector a, take its endpoint as the initial point of vector b, and complete the parallelogram. The area of this paralellogram taken with a certain sign is I(a conjugate)b. On what geometrical feature does the sign depend?

## The Attempt at a Solution

My initial thought is that is has something to do with angles and arguements, but my problem is figuring out how the sign changes.

What is I(a conjugate)?

Office_Shredder said:
What is I(a conjugate)?

I stands for imaginary like how in a+ib a is the Real part of z and b is the Imaginary part of z. a conjugate is like how the conjugate of a+ib is a-ib.

Ah, I see now. Try putting a and b into polar form: rei*theta

and the result will fall out pretty fast

So you are thinking of these vectors as in the complex plane? That is, the vector <x, y> is represented by x+ iy?

In three dimensions, the area of the parallelogram formed by vectors $\vec{u}$ and $\vec{v}$ is $|\vec{u}\times\vec{v}|$ and, of course, you need the absolute value because the sign of the cross product depends on the order of the vectors NOT on any property of the parallelogram. Looks to me like the same thing happens here: How does (conjugate a)b is differ from (conjugate b)a?

I realized my problem. If I considered the angle $$\beta$$-$$\alpha$$, then I was fine.

## 1. What are vectors in complex geometry?

Vectors in complex geometry are mathematical entities that represent both magnitude and direction. They are often represented by an arrow or line segment and can be added, subtracted, and multiplied by a scalar. In complex geometry, vectors are used to represent translations and rotations in the complex plane.

## 2. How do you perform vector addition and subtraction in complex geometry?

To perform vector addition and subtraction in complex geometry, you simply add or subtract the real and imaginary components of the vectors separately. For example, to add two vectors A and B, you would add the real components of A and B together and then add the imaginary components of A and B together to get the resulting vector. The same applies for subtraction.

## 3. What is a parallelogram in complex geometry?

A parallelogram in complex geometry is a quadrilateral with two pairs of parallel sides. In the complex plane, a parallelogram can be defined by the vertices of the shape, which are represented by complex numbers. The area of a parallelogram in complex geometry can be calculated using the cross product of two adjacent sides.

## 4. How do you determine if two vectors are parallel in complex geometry?

Two vectors in complex geometry are parallel if they have the same direction, which means they are scalar multiples of each other. This means that one vector can be obtained by multiplying the other vector by a complex number. To determine if two vectors are parallel, you can take the ratio of their real or imaginary components to see if they are equal.

## 5. How do you find the magnitude and direction of a vector in complex geometry?

The magnitude of a vector in complex geometry is calculated using the Pythagorean theorem, where the real and imaginary components of the vector are the legs of a right triangle. The direction of a vector in complex geometry is given by its argument, which is the angle between the vector and the positive real axis. This can be calculated using trigonometric functions such as tangent or arctangent.

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