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# Vector

A vector has a magnitude and direction. The length of the line shows its magnitude and the arrowhead points in the direction. We can add two vectors by joining them head-to-tail. And it doesn't matter which order we add them, we get the same result.

![](/files/-MPCaK9N8yuQv3Tgoxq1)

We can also subtract one vector from another. First, we reverse the direction of the vector we want to subtract, then add them as usual.

![](/files/-MPCaemAHzN8zclZRqNQ)

## **Adding Vector**

We can add vectors by adding the x parts and adding the y parts**.**

![](/files/-MPCbWTd4LbsOnpHWW5x)

$$
a = (8,13),
b = (26,7),
c = (8, 13) + (26, 7) = (8+26, 13+7) = (34, 20)
$$

## **Magnitude of a Vector**

$$
|a| = \sqrt( x^2 + y^2 )
$$

Magnitude of the vector,$$|b| = (6,8) = \sqrt( 6^2 + 8^2) = \sqrt( 36+64) = \sqrt100 = 10$$ 

## **Multiplying a Vector by a Vector**

* **Dot Product - Result is a Scaler**
  * $$a \cdot b = \lvert a \lvert \times \lvert b \lvert \times cos (\theta)$$ 
  * Multiply the length of a times the length of b, then multiply by the cosine of the angle between a and b.
  * Or, we can use the formula $$a \cdot b = a\_x \times b\_x + a\_y \times b\_y$$ 
  * Multiply the x's, multiply the y's, then add.
* **Cross Product - Results a Vector**
  * Cross Product a × b of two vectors is another vector that is at right angles to both.
  * $$a \times b = \lvert a \lvert \times \lvert b \lvert \times sin (\theta) \times n$$ 

## Polar and Cartesian Coordinates

![](/files/-MPCl7iS93Cv35SvmBXR)

![](/files/-MPClhBo1W9T-3gTpaJ2)

{% embed url="<https://www.mathsisfun.com/algebra/vectors.html>" %}

{% embed url="<https://www.mathsisfun.com/algebra/vectors-dot-product.html>" %}

{% embed url="<https://www.mathsisfun.com/algebra/vectors-cross-product.html>" %}