292 CHAPTER 14. VECTOR PRODUCTS

10. A girl drags a sled for 200 feet along the ground by pulling on a rope which is 30degrees from the horizontal with a force of 20 pounds. How much work does thisforce do?

11. 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?

12. An object moves 10 meters in the direction of j. There are two forces acting on thisobject F 1 = i+ j+k, 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?

13. An object moves 10 meters in the direction of j+ i. There are two forces acting onthis object F 1 = i+j+2k, and F 2 = 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?

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

15. Show that (a ·b) = 14

[|a+b|2 −|a−b|2

].

16. Prove from the axioms of the dot product the parallelogram identity which assertsthat |a+b|2 + |a−b|2 = 2 |a|2 +2 |b|2 .

17. Suppose f ,g are two continuous functions defined on [0,1] . Define

( f ·g) =∫ 1

0f (x)g(x)dx.

Show this dot product satisfies conditions 14.1 - 14.5. Explain why the CauchySchwarz inequality continues to hold in this context and state the Cauchy Schwarzinequality in terms of integrals.

14.4 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 14.4.1 Three vectors a,b,c form a right handed system if when youextend the fingers of your right hand along the vector a and close them in the direction ofb, the thumb points roughly in the direction of c.

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