2. Vectors: 5 Minute Review Vectors can be added or subtracted. ◦ To add vectors graphically, draw one after the other, tip to tail. ◦ To add vectors algebraically,  Resolve the vectors into components.  Add the components of each direction.  Use the Pythagorean Theorem to find the magnitude of the resultant vector and inverse tan to find the angle.

3. Vector Addition Example A boy pushes horizontally on a wagon with a force of 10 N. A girl pulls on the wagon’s handle at an angle of 30 degrees from the horizontal with a force of 8 N. What is the net force acting on the wagon?

4. Unit Vectors A unit vector is a vector that points along the x, y or z axis and is one unit long. ◦ The symbols for the x, y, and z unit vectors are ◦ Any vector can be expressed as a sum of its x, y and z components multiplied by unit vectors. ◦ The dimensions of the quantity are stated along with the unit vectors, e.g.

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6. Unit Vector Example A car has a velocity of 6 m/s in a direction 30 o north of east. Express this vector in terms of unit vectors. Let east be the positive x direction and north the positive y direction.

7. Operations with Vectors: Magnitude To find the magnitude of a vector, use the Pythagorean Theorem.

8. Operations with Vectors: Addition and subtraction To add two vectors, add their components.

9. Vector Addition & Subtracting Graphical method Addition: Connect head to tail Subtraction: flip the subtrahend 180°, then connect head to tail.

10. Multiplication of Vectors: 1.Vector Product of a Scalar and a Vector 2.Scalar Product of Two Vectors 3.Vector Product of Two Vectors

11. Multiplication of Vector by Scalar Applications momentump = mv electric forceF = qE Result A vector with the same direction, a different magnitude and perhaps different units.

12. Multiplication of Vector by Vector (Dot Product) Application workW = F  d Result A scalar with magnitude and no direction.

13. Multiplication of Vector by Vector (Dot Product) C = A  B C = AB cos  A B 

14. Dot Product Practice i i = i j = i k = j j = j k = j i = k i = k k = k j = 1x1x cos0 1x1x cos 90 5i 3i = 4i 2j = -4i 2k = 5j -3j = 2j 2 k = 1j 9 i = 9k 7 i = -√3k -2k = ½k √2j =

15. Multiplication of Vector by Vector (Cross Product) Application Work  = r  F Magnetic force F = qv  B Result A vector with magnitude and a direction perpendicular to the plane established by the other two vectors.

16. Multiplication of Vector by Vector (Cross Product) C = A  B C = AB sin  (magnitude) Direction determined by Right Hand Rule A B 

17. Multiplication of Vector by Vector (Cross Product) C = A  B A B 

18. Operations with Vectors: The Cross Product The cross product yields a vector at right angles to the two vectors.

19. Cross Product Practice i x i = i x j = i x k = j x j = j x k = j x i = k x i = k x k = k x j = 1x1x sin0 1x1x sin90 1x1x sin0 1x1x sin90 5i x4i = 3i x 3j = -9i x 6k = 5j x5 j = √3j x √3k = 2.4j x 2 i = ½k x ¾i = -2k x 2k = -2k x 2j =

20. Cross Product Practice i x i = i x j = i x k = j x j = j x k = j x i = k x i = k x k = k x j = 1x1x sin0 1x1x sin90 1x1x sin0 1x1x sin90 5i x4i = 3i x 3j = -9i x 6k = 5j x5 j = √3j x √3k = 2.4j x 2 i = ½k x ¾i = -2k x 2k = -2k x 2j =

21. Vector Multiplication: 2 types: Cross product & dot product