8.5 Vectors!

So, this is a conversation I’m anticipating: You: What in the world is a vector, Ms T? Me: I’m glad you asked. Me: A vector is any quantity with magnitude and direction. You: ???? You: Thanks, but not particularly helpful.

So, how about this? A vector describes where something is headed and how far it will get. Let’s start with a graph and work from there: Magnitude of vector: Distance from initial point K (0,0) to terminal point W (2,3) (in this example) The vector can be described using the ordered pair of the terminal point: ‹2,3›

How do you calculate magnitude? Remember the work we’ve done with trigonometry? Yep, that’s how. Or, if you don’t know the angle measures, perhaps you will have enough information to use Pythagorean Theorem. Let’s see an example…

How does direction work, then? Direction is measured using the cardinal points -North, South, East, West Using trigonometry (oh no, not that again), you can find the angle made by the vector and either the x or y axis. The vertex will be at the origin.

Let’s brush up on Cardinal Points

Naming Vectors A vector can be named using one lowercase letter with an arrow above it. For example: a So, to be clear about notation: a (just a random line) a (vector)

Adding vectors together You can add two vectors together. The outcome is called the resultant. So, if you have two vectors ‹x 1, y 1 › and ‹x 2, y 2 ›, the resultant will look like this: ‹x 1 + x 2, y 1 + y 2 › Let’s look at an example…

Adding Vectors: ‹4, 2› and ‹2, 5›: ‹4 + 2, 2 + 5› = ‹6, 7› How does it look on a graph?

Here’s something to help you remember the two aspects of a vector:

Examples! A ship leaves the dock and ends up 120 miles east and 45 miles south. It travels in a straight path. a) What is the direction of the vector that represents the ship’s trip? Round to the nearest tenth. b) What is the magnitude of the vector that represents the ship’s trip? (Use trigonometry, not Pythagorean theorem.) Round to the nearest whole number.

Examples! Kennedy goes on a run and ends up 2 miles west and 3 miles south of MIHS. She runs in a straight path. a) What is the direction of the vector that represents her run? Round to the nearest tenth. b) What is the magnitude of the vector that represents her run? (Use trigonometry, not Pythagorean theorem.) Round to the nearest whole number.

So, things to remember: SOH CAH TOA Draw the picture! Vectors always start at the origin Vectors are the hypotenuse of a right triangle made with part of the x-axis and part of the y-axis Vectors include both magnitude and direction – How far does it travel? – Where is it going?