Vectors and Vector Multiplication

1. Vectors and Vector Multiplication

2. Vector quantities are those that have magnitude and direction, such as: • Displacement, x or • Velocity, • Acceleration, • Force, • Torque, • Electric field, ….to name just a few

3. Scalar quantities have only magnitude: • Speed, v • Distance, d • Time, t • Energy, E • Power, P • Charge, q • Electric potential, V

4. Multiplication of scalar quantities follows all the “usual” rules, including:Distributive a(b+c) = ab + acCommutative ab = baAssociative (ab)c = a(bc)

5. Addition of scalars follows these properties:Commutative a+b = b+aAssociative (a+b)+c = a+(b+c)Subtraction a+(-b) = a-b

6. Addition of vectors is commutative and associative and follows the subtraction rule:A+B = B+A(A+B)+C = A+(B+C)A-B = A+(-B)

7. A+B = B+A B A A+B B+A

8. (A+B)+C = A+(B+C) B A C A+B B+C (A+B)+C A+(B+C)

9. A-B = A+(-B) B -B A A-B

10. Multiplication of a scalar and a vector follows previous rules:aB = Baa(B+C) = aB + aC

11. However, multiplication of vectors has a new set of rules—the vector cross product (or “vector product”) and the vector dot product or “scalar product”.

12. Vector Dot Productor Scalar ProductA·B = AB cosEssentially, this means multiplying the first vector times the component of the second vector that is in the same direction as the first vector—yielding a product that is a scalar quantity.

13. B B sin A  B cos A·B = AB cos Multiple the magnitude of vector A times the magnitude of vector B times the cosine of the angle between them—or multiply the components that are in the same direction. The answer is a scalar with the units appropriate to the product AB.

14. Vector Cross Productor Vector ProductAxB = AB sinEssentially, this means multiplying the first vector times the component of the second vector that is perpendicular to the first vector—yielding a product that is a vector quantity. The direction of the new vector is found using the right hand rule.

15. B B sin A  B cos Multiple the magnitude of vector A times the magnitude of vector B times the sine of the angle between them—or multiply the components that are perpendicular. The answer is a vector with the units appropriate to the product AB and direction found by using the right hand rule.

16. For example, let’s take the vector cross product: F = q (vxB) where q is the charge on a proton, v is 3x105 m/s to the left on the paper, and B is 500 N/C outward from the paper toward you. The equation for this is also: F = qvB sin

17. The answer for the force is 2.4 x 10-11 newtons toward the top of the paper.

18. Unit vectors Unit vectors have a size of “1” but also have a direction that gives meaning to a vector. We use the “hat” symbol above a unit vector to indicate that it is a unit vector. For example, is a vector that is 1 unit in the x-direction. The quantity 6 meters is a vector 6 meters long in the x-direction.

19. Did you realize that you have been using a right-handed Cartesian coordinate system in mathematics all these years?

20. You can check your use of the right hand rule, because

21. Here are a few for practice:

22. We can also do dot products with unit vectors. Try these: 1 0 0 1 8 12 m2

23. The calculation of work is a scalar product or dot product: What is the work done by a force of 6 newtons east on an object that is displaced 2 meters east? What is the work done by a force of 6 newtons east on an object that is displaced 2 meters north? What is the work done by a force of 6 newtons east on an object that is displaced 2 meters at 30 degrees north of east? 12 joules zero 10.4 joules

24. In summary: • In an equation or operation with a scalar or dot product, the answer is a scalar quantity that is the product of two vectors. • The dot product is found by multiplying the components of vectors that are in the same direction. • In an equation or operation with a vector or cross product, the answer is a vector quantity that is the product of two vectors. • The cross product is found by multiplying the components of vectors that are perpendicular to each other.