Vectors – A Super Brief Over View and An Intro to the Calculus of Them

1. We first learned about derivatives and then explored their practical applications with motion along a line (position, velocity, and acceleration). Now… It is time to see how our knowledge of the calculus of parametric equations can be used to describe motion along a curve (any direction we want!!) To do this, we will use VECTORS!!!

2. Vectors – A Super Brief Over View and An Intro to the Calculus of Them Vector – A directed line segment that represents any quantity that has magnitude and direction.

3. Where have we seen these before? • Let’s return to the b-ball picture. • a.) What was a position vector in this case? • b.) What was a velocity vector? • c.) How about an acceleration vector? • d.) What do each of these vectors tell us?

4. Velocity and Acceleration Vectors… Oh, and Speed too!!! A Position Vector describes the direction and distance a point/object is from where it started (think basketball starting point and an arrow meeting its current position) A Velocity Vector describes the direction in and speed with which a point/object is traveling. An Acceleration Vector describes the direction in and rate at which a point/object’s velocity is changing.

5. Before we find position, velocity and acceleration vectors…let’s get warmed up graphing Vector-Valued Functions.

6. Vector-Valued Functions Vector – Valued Function – A function whose graph is traced out by vectors (but really acts like a parametric curve!) Let’s trace out a general curve!

7. Sketch r(t) = • Just like parametric • Remember Orientation

8. Curves in Space (3D!) Sketch Point being…. WE SKETCH VECTOR-VALUED FUNCTIONS THE SAME AS IF THEY WERE P-METRIC!!

9. Velocity Vector or Tangent Vector The velocity or tangent vector represents the instantaneous rate of change of an object at any specific value of t. NOTE! FAMILIAR?

10. It follows the acceleration vector is: Since you have a velocity vector, by finding its magnitude, you obtain _____________ at a point. speed Remind you of anything? Let’s check out the conceptual!

11. A true-to-form FR from an example in our text: Find the velocity vector, speed, and acceleration vector of a particle that moves along the following Let’s check out this problem graphically and evaluate each of the following:

12. a.) Sketch the path of a particle moving along the above position vector. b.) Sketch the velocity and acceleration vector at (3,3).

13. a.) Sketch the path of a particle moving along the above position vector. b.) Sketch the velocity and acceleration vector at (1,1).

14. A particle moves along a curve described by the equations given below. Find: a.) The velocity vector at t = 2 b.) The speed of the particle at t = 2 c.) The acceleration vector at t = 2. d.) The angle the particle is moving relative to the x-axis at t = 2

15. A particle moves along a line whose equation is 1.) Write an equation for the position vector. 2.) Write the velocity and acceleration vectors. 3.) find the speed of the object.

18. A quick trip back to p-metrics with a little vectorage! a.) The particle’s position at t = 3 b.) The speed of the particle at t = 3 c.) Direction of the particle at t = 3

19. Projectile Motion – Vectorizing what we already know!!!

20. The QB of a football team releases a pass at a height of 7 feet above the playing field, and the football is caught by a receiver 30 yards directly downfield at a height of 4 feet. The pass is released at an angle of 35 degrees with the horizontal. a.) Find the speed of the football when it is released. b.) Find the maximum height of the football. c.) Find the time the receiver has to reach the proper reception position after the QB releases the football. TREAT LIKE A P-METRIC PROBLEM!!!