84 CHAPTER 6. VECTOR PRODUCTS

Definition 6.2.18 Suppose you have a vector space, V and for z,w ∈ V and α a scalara norm is a way of measuring distance or magnitude which satisfies the properties 6.16 -6.18. Thus a norm is something which does the following.

||z|| ≥ 0 and ||z||= 0 if and only if z = 0 (6.19)

If α is a scalar, ||αz||= |α| ||z|| (6.20)

||z+w|| ≤ ||z||+ ||w|| . (6.21)

Here is understood that for all z ∈V, ||z|| ∈ [0,∞).

6.3 Exercises1. Find (1,2,3,4) · (2,0,1,3) .

2. Use formula 6.12 to verify the Cauchy Schwartz inequality and to show that equalityoccurs if and only if one of the vectors is a scalar multiple of the other.

3. 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 the Cauchy Schwarzinequality |u∗v| ≤ (u∗u)1/2 (v ∗v)1/2 . Hint: Do not try to do this with methodsfrom trigonometry.

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

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

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

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

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

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

10. 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 thetwo vectors F and D. |F |cosθ is sometimes called the component of the force, Fin the direction, D.

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?