# Algebra/Vectors in two dimensions

Imagine an arrow that has its tail at the origin <0,0> (the carrots denote a vector) and its head at some coordinate <x,y>. These coordinates are known as the x-component and y-component, respectively. This is a vector: it has a magnitude and direction:

**Magnitude**is simply the length of the vector as defined by Pythagorean's Theorem. Hence, the length of any vector is simply**√(x**^{2}+ y^{2})**Direction**is the angle the vector is pointing. If the vector is pointing along the positive x-axis, then the angle is 0˚ (0 radians). If it is pointing along the negative x-axis, then the angle is 180˚ (π radians). If it points along the positive y-axis, the angle is 90˚ (π/2 radians), and if it points along the negative y-axis, the angle is 270˚ (3π/2 radians).

- Indented line *Because of trigonometry, we can figure out that x

Following this reasoning, the angle along the positive x-axis is also 360˚ (2π radians), and the angle along the positive y-axis is also 450˚ (5π/2 radians), etc. Similarly, the angle along the negative y-axis is -90˚ (-π/2 radians) and the angle along the negative x-axis is -180˚ (-π radians), etc. Calculating direction is more difficult than calculating magnitude. More on this in a moment.

Interestingly, the x and y components can be calculated by knowing the magnitude, direction, and some trigonometric identities.

**x = (Magnitude) • cos (Direction)****y = (Magnitude) • sin (Direction)**

Returning to direction: given the above equations and the equation for magnitude, we can solve for a vectors direction mathematically by solving the system:

- (Magnitude) = √(x
^{2}+ y^{2}) - x = (Magnitude) • cos (Direction)
- y = (Magnitude) • sin (Direction)

Hence,

- x = √(x
^{2}+ y^{2}) • cos (Direction) - y = √(x
^{2}+ y^{2}) • sin (Direction)

Therefore,

- x/√(x
^{2}+ y^{2}) = cos (Direction) - y/√(x
^{2}+ y^{2}) = sin (Direction)

Finally,

- Direction = arccos(x/√(x
^{2}+ y^{2})) = arcsin(y/√(x^{2})

### Calculating Direction - Example[edit | edit source]

Let's say we have a vector <√3,1> and we wish to find the direction of it. First, we calculate the magnitude:

- (Magnitude) = √(x
^{2}+ y^{2}) - (Magnitude) = √((√3)
^{2}+ (1)^{2}) - (Magnitude) = √(3 + 1)
- (Magnitude) = √(4)
- (Magnitude) = 2

Then, we can calculate direction:

- Direction = arccos(x/√(x
^{2}+ y^{2})) = arcsin(y/√(x^{2}) - Direction = arccos(√(3)/2) = arcsin(1/2)

Using degrees and the *arccos* equation, we find that direction can be 30˚, or 330˚. Using the *arcsin* equation, we find that direction can be either 30˚ or 150˚. The common direction is 30˚, so we know this is the direction of the vector.
**Vector Addition**
Like numbers (scalars), vectors can be added and subtracted. For vectors you simply add each of the component parts. Subtraction works the same way Hence:

- <5,2> + <-7,4> = <-2,6>
- <5,2> - <-7,4> = <12,-2>

### Scalar Multiplication[edit | edit source]

If you multiply a scalar by a vector, you simply multiply each of the components by that scalar:

- 3•<5,2> = <15,6>
- -2•<-7,4> = <14,-8>

### Dot Product[edit | edit source]

You can't multiply two two-dimensional vectors together, but you can take the **dot product**. To do this, you take the sum of each of the corresponding components' products:

- <5,2>•<-7,4> = (-35) + (8) = -27
- <7,4>•<3,8> = (21) + (32) = 53

It turns out that this has numerous applications.