# 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.

![](https://2552912007-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LzGBVuquaFNrwdmJna0%2F-MPC_WPm3OPkp-BOFAUO%2F-MPCaK9N8yuQv3Tgoxq1%2FScreen%20Shot%202020-12-22%20at%208.16.25%20PM.png?alt=media\&token=cae20931-bab4-40cd-b9e3-86936f0362c0)

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.

![](https://2552912007-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LzGBVuquaFNrwdmJna0%2F-MPCaP9BypH5okm-8lVm%2F-MPCaemAHzN8zclZRqNQ%2FScreen%20Shot%202020-12-22%20at%208.18.04%20PM.png?alt=media\&token=bba67ce3-7d93-4078-8e45-f4c0333a1761)

## **Adding Vector**

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

![](https://2552912007-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LzGBVuquaFNrwdmJna0%2F-MPCaP9BypH5okm-8lVm%2F-MPCbWTd4LbsOnpHWW5x%2FScreen%20Shot%202020-12-22%20at%208.21.40%20PM.png?alt=media\&token=5c77f2ee-096b-47e3-be80-35e20af797d2)

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

## **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

![](https://2552912007-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LzGBVuquaFNrwdmJna0%2F-MPCk9z4KGeIGJu5V-1v%2F-MPCl7iS93Cv35SvmBXR%2FScreen%20Shot%202020-12-22%20at%209.03.47%20PM.png?alt=media\&token=859766bd-0c70-4aa1-be85-ae94c6ce5106)

![](https://2552912007-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LzGBVuquaFNrwdmJna0%2F-MPCk9z4KGeIGJu5V-1v%2F-MPClhBo1W9T-3gTpaJ2%2FScreen%20Shot%202020-12-22%20at%209.04.21%20PM.png?alt=media\&token=8a25a1ee-902a-4994-ba40-6b18fcc13df4)

{% 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>" %}