40 CHAPTER 3. VECTOR PRODUCTS

3.3 Exercises1. Use formula 3.11 to verify the Cauchy Schwarz inequality and to show that equality

occurs if and only if one of the vectors is a scalar multiple of the other.

2. For u,v vectors in R3, define the product, u∗v ≡ u1v1 + 2u2v2 + 3u3v3. Show theaxioms for a dot product all hold for this funny product. Prove

|u∗v| ≤ (u∗u)1/2 (v∗v)1/2 .

Hint: Do not try to do this with methods from trigonometry.

3. Find the angle between the vectors 3i− j−k and i+4j+2k.

4. Find the angle between the vectors i−2j+k and i+2j−7k.

5. Find proju (v) where v =(1,0,−2) and u =(1,2,3) .

6. Find proju (v) where v =(1,2,−2) and u =(1,0,3) .

7. Find proju (v) where v =(1,2,−2,1) and u =(1,2,3,0) .

8. Does it make sense to speak of proj0 (v)?

9. If F is a force and D is a vector, show projD (F) = (|F|cosθ)u where u is the unitvector in the direction of D, u = D/ |D| and θ is the included angle between the twovectors, F and D. |F|cosθ is sometimes called the component of the force, F in thedirection, D.

10. Prove the Cauchy Schwarz inequality in Rn as follows. For u,v vectors, consider

(u−projv u) ·(u−projv u)≥ 0

Now simplify using the axioms of the dot product and then put in the formula for theprojection. Of course this expression equals 0 and you get equality in the CauchySchwarz inequality if and only if u = projv u. What is the geometric meaning ofu = projv u?

11. A boy drags a sled for 100 feet along the ground by pulling on a rope which is 20degrees from the horizontal with a force of 40 pounds. How much work does thisforce do?

12. 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?

13. A large dog drags a sled for 300 feet along the ground by pulling on a rope which is45 degrees from the horizontal with a force of 20 pounds. How much work does thisforce do?

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?