1. Vectors & Trigonometric Form By: Ina Salvaleon, Hannah Galt, Irem Guney

2. Topics Properties of vectors Operations of vectors Changing to standard form Applications of vectors Changing to trigonometric form

3. Formulas– Vectors Component form: Magnitude and direction: Angle between two vectors: Unit vector of v: Dot product: Linear combination form: Projection vector: Work:

4. Vectors u= v= Find exact magnitude: Find direction angle of v: arctan (6/-3) + 180° = 116.57°

5. Vectors u= v= Vector parallel to u: Vector orthogonal to v: switch and change one sign

6. Vectors u= v= Find a unit vector in the same direction as u: / = Find a unit vector in the same direction as u, but with a magnitude of 7: (7) =

7. Vectors u= v= Find the dot product of u and v: x 1 x 2 +y 1 y 2 = (5)(-3)+(-4)(6) = -39 Find the angle between u and v:

8. Vectors u= v= Find the projection of u onto v: (-39/45) = = Component of u that is orthogonal to proj v u: w 2 = u-w 1 = - =

9. Vectors Change 6-3i to trig form (radians): Change to standard form:

10. Application of Vectors A wagon full of children with a combined weight of 110 pounds is on a hill with an incline of 9 degrees. What force will someone have to exert to keep it from rolling down the hill? What is the force perpendicular to the hill? A: 17.2 poundsB:108.6 pounds

11. Application of Vectors Three vectors with magnitudes of 150, 200, and 100 act on an object at angles of 40°,-80°, and 120 °, respectively with the positive –axis. What is the resultant vector? What is its magnitude and direction? A:B:

12. Application of Vectors A tractor pulls a log 800 meters and the tension in the cable connecting the tractor and the log is approximately 14,000 newtons. What is the work done if the direction of the force is parallel to the horizontal? What is the work done if the direction of the force is 20° above the horizontal? A:B:

13. Formulas– Trigonometric Form Trig form: Standard form: Multiplying: Dividing: Exponents: Roots:

14. Multiplying in Trig Form 1.Simplify to 2.Multiply 7 times 8 3.Add and Solution:

15. Dividing in Trig Form 1.Simplify to 2.Divide 12 by 3 3.Subtract by Solution:

16. Powers of Complex Numbers 1.Find the radius and angle. 2.Plug in the radius and angle into the powers formula.

17. Powers of Complex Numbers Solution in trig (radians) and standard forms:

18. Roots of Complex Numbers Find the 5 th roots of 1.Find the radius and angle. 2.Plug in the radius and angle into the roots formula.

19. Roots of Complex Numbers Find the 5 th roots of 3.Plug in the values of 0, 1, 2, 3, & 4 for k. Make sure your calculator is in radians!

20. Roots of Complex Numbers Find the 5 th roots of Solution in trig (radians) and standard forms:

21. Homework Ch. 6 Review (pg. 480) 6.3: #43, 59, 71 6.4: #77, 85, 97(just part a, projection of u onto v) 6.5: #107, 115, 119