- Find ⋅.
- Use formula 3.12 to verify the Cauchy Schwartz inequality and to show that equality occurs if and only if one of the vectors is a scalar multiple of the other.
- For u,v vectors in ℝ
^{3}, define the product u ∗ v ≡ u_{1}v_{1}+ 2u_{2}v_{2}+ 3u_{3}v_{3}. Show the axioms for a dot product all hold for this funny product. Prove≤^{1∕2}^{1∕2}. Hint: Do not try to do this with methods from trigonometry. - Find the angle between the vectors 3i − j − k and i + 4j + 2k.
- Find the angle between the vectors i − 2j + k and i + 2j − 7k.
- Find proj
_{u}where v =and u =. - Find proj
_{u}where v =and u =. - Find proj
_{u}where v =and u =. - Does it make sense to speak of proj
_{0}? - If F is a force and D is a vector, show proj
_{D}=u where u is the unit vector in the direction of D, u = D∕and θ is the included angle between the two vectors F and D.cos θ is sometimes called the component of the force, F in the direction, D. - A boy drags a sled for 100 feet along the ground by pulling on a rope which is 20 degrees from the horizontal with a force of 40 pounds. How much work does this force do?
- A girl drags a sled for 200 feet along the ground by pulling on a rope which is 30 degrees from the horizontal with a force of 20 pounds. How much work does this force do?
- A large dog drags a sled for 300 feet along the ground by pulling on a rope which is 45 degrees from the horizontal with a force of 20 pounds. How much work does this force do?
- How much work in Newton meters does it take to slide a crate 20 meters along a loading dock
by pulling on it with a 200 Newton force at an angle of 30
^{∘}from the horizontal? - An object moves 10 meters in the direction of j. There are two forces acting on this object
F
_{1}= i + j + 2k, and F_{2}= −5i + 2j−6k. Find the total work done on the object by the two forces. Hint: You can take the work done by the resultant of the two forces or you can add the work done by each force. Why? - An object moves 10 meters in the direction of j + i. There are two forces acting on this object
F
_{1}= i + 2j + 2k, and F_{2}= 5i + 2j−6k. Find the total work done on the object by the two forces. Hint: You can take the work done by the resultant of the two forces or you can add the work done by each force. Why? - An object moves 20 meters in the direction of k + j. There are two forces acting on this object
F
_{1}= i+j+2k, and F_{2}= i+2j−6k. Find the total work done on the object by the two forces. Hint: You can take the work done by the resultant of the two forces or you can add the work done by each force. - If a,b, and c are vectors. Show that
_{⊥}= b_{⊥}+ c_{⊥}where b_{⊥}= b−proj_{a}. - In the discussion of the reflecting mirror which directs all rays to a particular point . Show that for any choice of positive C this point is the focus of the parabola and the directrix is y = p −.
- Suppose you wanted to make a solar powered oven to cook food. Are there reasons for using a mirror which is not parabolic? Also describe how you would design a good flash light with a beam which does not spread out too quickly.
- Show that =.
- Prove from the axioms of the dot product the parallelogram identity
^{2}+^{2}= 2^{2}+ 2^{2}. - Suppose f,g are two continuous functions defined on . Define= ∫
_{0}^{1}fgdx. Show this dot product satisfies conditions 3.1 - 3.5. Explain why the Cauchy Schwarz inequality continues to hold in this context and state the Cauchy Schwarz inequality in terms of integrals.

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