10.3 Vector Valued Functions Greg Kelly, Hanford High School, Richland, Washington.

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1 10.3 Vector Valued Functions Greg Kelly, Hanford High School, Richland, Washington

2 Any vector can be written as a linear combination of two standard unit vectors. The vector v is a linear combination of the vectors i and j. The scalar a is the horizontal component of v and the scalar b is the vertical component of v.

3 We can describe the position of a moving particle by a vector, r ( t ). If we separate r ( t ) into horizontal and vertical components, we can express r ( t ) as a linear combination of standard unit vectors i and j.

4 In three dimensions the component form becomes:

5 Graph on the TI-89 using the parametric mode. MODE Graph…….2 ENTER Y= ENTER WINDOW GRAPH

6 Graph on the TI-89 using the parametric mode. MODE Graph…….2 ENTER Y= ENTER WINDOW GRAPH

7 Most of the rules for the calculus of vectors are the same as we have used, except: “Absolute value” means “distance from the origin” so we must use the Pythagorean theorem. Note: The magnitude of the velocity is by definition identical to the speed which is a scalar (not a vector) and never negative; however, velocity is a vector because it has direction and magnitude

8 Example 5: a) Find the velocity and acceleration vectors. b) Find the velocity, acceleration, speed and direction of motion at.

9 Example 5: b) Find the velocity, acceleration, speed and direction of motion at. velocity: acceleration:

10 Example 5: b) Find the velocity, acceleration, speed and direction of motion at. speed: direction:

11 Example 6: a) Write the equation of the tangent line where. At : position: Slope= To write equation:

12 The horizontal component of the velocity is. Example 6: b) Find the coordinates of each point on the path where the horizontal component of the velocity is 0. 