3.4. THE CROSS PRODUCT 41

15. An object moves 10 meters in the direction of j. There are two forces acting onthis object, F1 = i+ j+2k, and F2 =−5i+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. Why?

16. An object moves 10 meters in the direction of j+ i. There are two forces acting onthis object, F1 = i+2j+2k, and F2 = 5i+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. Why?

17. An object moves 20 meters in the direction of k+ j. There are two forces acting onthis object, F1 = i+ j+ 2k, and F2 = 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, c are vectors. Show that (b+ c)⊥ = b⊥+ c⊥ where b⊥ = b−proja (b) .

19. Find (1,2,3,4) · (2,0,1,3) .

20. Show that (a ·b) = 14

[|a+b|2−|a−b|2

].

21. Prove from the axioms of the dot product the parallelogram identity, |a+b|2 +|a−b|2 = 2 |a|2 +2 |b|2 .

22. Recall that the open ball having center at a and radius r is given by

B(a,r)≡ {x : |x−a|< r}

Show that if y ∈ B(a,r) , then there exists a positive number δ such that B(y,δ ) ⊆B(a,r) . (The symbol⊆means that every point in B(y,δ ) is also in B(a,r) . In words,it states that B(y,δ ) is contained in B(a,r) . The statement y ∈ B(a,r) says that y isone of the points of B(a,r) .) When you have done this, you will have shown thatan open ball is open. This is a fantastically important observation although its majorimplications will not be explored very much in this book.

3.4 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 3.4.1 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.

For an example of a right handed system of vectors, see the following picture.