6.4. THE CROSS PRODUCT 85

14. How much work in Newton meters does it take to slide a crate 20 meters along aloading dock by pulling on it with a 200 Newton force at an angle of 30◦ from thehorizontal?

15. An object moves 10 meters in the direction of j. There are two forces acting on thisobject F 1 = i+ j+2k, and F 2 = −5i+2 j−6k. Find the total work done on theobject by the two forces. Hint: You can take the work done by the resultant of thetwo forces or you can add the work done by each force. Why?

16. An object moves 10 meters in the direction of j+ i. There are two forces acting onthis object F 1 = i+ 2j+ 2k, and F 2 = 5i+ 2 j− 6k. Find the total work done onthe object by the two forces. Hint: You can take the work done by the resultant ofthe two forces or you can add the work done by each force. Why?

17. An object moves 20 meters in the direction of k+j. There are two forces acting onthis object F 1 = i+j+2k, and F 2 = i+2j−6k. Find the total work done on theobject by the two forces. Hint: You can take the work done by the resultant of thetwo forces or you can add the work done by each force.

18. If a,b, and c are vectors. Show that (b+c)⊥ = b⊥+c⊥ where b⊥ = b−proja (b) .

19. In the discussion of the reflecting mirror which directs all rays to a particular point(0, p) . Show that for any choice of positive C this point is the focus of the parabolaand the directrix is y = p− 1

C .

20. Suppose you wanted to make a solar powered oven to cook food. Are there reasonsfor using a mirror which is not parabolic? Also describe how you would design agood flash light with a beam which does not spread out too quickly.

21. Show that (a ·b) = 14

[|a+b|2−|a−b|2

].

22. Prove from the axioms of the dot product the parallelogram identity which is thefollowing: |a+b|2 + |a−b|2 = 2 |a|2 +2 |b|2 .

23. Suppose f ,g are two continuous functions defined on [0,1] . Define the inner product( f ·g) =

∫ 10 f (x)g(x)dx. Show this dot product satisfies conditions 6.1 - 6.5. Explain

why the Cauchy Schwarz inequality continues to hold in this context and state theCauchy Schwarz inequality in terms of integrals.

6.4 The Cross ProductThe cross product is the other way of multiplying two vectors in R3. It is very differentfrom the dot product in many ways. First the geometric meaning is discussed and thena description in terms of coordinates is given. Both descriptions of the cross product areimportant. The geometric description is essential in order to understand the applicationsto physics and geometry while the coordinate description is the only way to practicallycompute the cross product.

Definition 6.4.1 Three vectors a,b,c form a right handed system if when you extend thefingers of your right hand along the vector a and close them in the direction of b, the thumbpoints roughly in the direction of c.