1. 8.5.1 – Vectors in the Cartesian Plane

2. The Basics Throughout math and physics, many things may be influenced by more than just a direction or length (magnitude) For those that are influenced by two factors, such as force OR velocity (magnitude AND direction), call them vectors

3. Terminology Vectors will usually be denoted with {a,b} as opposed to using parenthesis Initial Point = starting point of vector in a plane Terminal Point = ending point of a vector in a plane Similar to line segments

4. Two vectors are considered equal ONLY if their length and direction are the same – Initial and Terminal points do not matter!

5. Magnitude and Addition The correctly measure the length of a vector, or the magnitude or norm of a vector, we use the notation and equation; For the vector u with initial and terminal point {a,b}; || u || =

6. For vector addition, with vectors u and v, we use the notation u + v Two ways to add vectors – 1) Graphically; match one vector with the terminal end to the initial end (nose to tail); draw the third segment – 2) Component form (adding together each individual element)

7. Example. Find u + v from the following diagram:

8. Example. Find 2u - v from the following diagram:

9. Example. Find u + 3v from the following diagram:

10. Component Form With component form, we can find the magnitude of vectors (or, do other operations) In component form, we will pull out two parts of information – 1) Horizontal portion of the slope (u 1 ) – 2) Vertical portion of the slope (u 2 ) – All based on initial and terminal point of the vector (order DOES matter)

11. Component Form Cont’d Given two vectors u = {u 1, u 2 } and v = {v 1, v 2 }; – u + v = {u 1 + v 1, u 2 + v 2 } – || u || = The magnitude is still the same as the distance formula; just now using “short hand” to write it

12. Example. Find the component form and magnitude of the following vectors; assume the first is the initial point and the second is the terminal point. (-3, 5); (3, 3)

13. Example. Find the component form and magnitude of the following vectors; assume the first is the initial point and the second is the terminal point. u = (1,0) v = (-1, -2)

14. Assignment Pg. 666 1-16 all