Exercises

(Note: Applets in the "Tools" webpage of each section are intended to enhance the exercises. Use these tools to check your work, enhance your understanding of the material, or to perform calculations)

Find the vector with initial point P₁ and final point P₂. Sketch the result in either the xy-plane or in R³, whichever is appropriate.  

1. P1( 2,3) , P2( 3,5) 2. P1( 3,7) , P2( -4,-11) 3. P1( 7,9,2) , P2( 3,7,0) 4. P1( 7,9,2) , P2( 3,7,0) 5. P1( -4,-7,1) , P2( -3,3,5) 6. P1( 0,0,0) , P2( 1,3,1) 7. P1( 8,-10,3) , P2( 1,10,7) 8. P1( -1,-3,-1) , P2( 0,0,0) 9. P1( a,b,c) , P2( a+1,b+2,c+3) 10. P1( a,2,c) , P2( a+1,5,c+3)

Find u+v and u-v for the given vectors u and v, and then sketch u, v, u+v and u-v. 

11. u = á 3,1 ñ , v = á2,5 ñ 12. u = á 1,3 ñ , v = á 2,5 ñ 13. u = á -1,-2 ñ , v = á-3,5 ñ 14. u = á 0,1 ñ , v = á 1,0 ñ 15. u = á -1,-1 ñ , v = á 1,1 ñ 16. u = á2,3 ñ , v = á 4,6 ñ 17. u = á 1,0,0 ñ , v = á 0,1,0 ñ 18. u = á0,1,0 ñ , v = á 0,0,1 ñ 19. u = á 2,1,0 ñ , v = á 0,3,5 ñ 20. u = á1,-2,3 ñ , v = á 2,5,-4 ñ

The following vectors represent forces acting on a single object in space. Draw the Force Diagram and determine the net force acting on the object. Is the object at equilibrium?

21. F1 = á 0,-9.8 ñ ,  F2 = á -3,3 ñ , F3 = á3,3 ñ 22. F1 = á 0,-98 ñ ,  F2 = á -30,30 ñ , F3 = á30,30 ñ 23. F1 = á 0,-9.8 ñ ,  F2 = á 0,9.8 ñ, F3 = -3i , F4 = 3i 24. F1 = á 0,-15 ñ ,  F2 = á 0,15 ñ 25. F1 = á 0,0,-9.8 ñ ,  F2 = á 10,0,10 ñ , F3 = á0,10,10 ñ 26. F1 = á 1,0,-1 ñ ,  F2 = á -1,1,0 ñ , F3 = á0,-1,1 ñ

27.  An airplane heads due north at 300 mph through a cross-wind blowing due east at 50 mph. What is the actual speed and direction of the airplane?

28. An airplane heads due east at 300 mph through a tailwind whose velocity is given by

w = á 20,20 ñ

How fast is the tailwind blowing? In what direction? How fast is the plane flying? In what direction?

29. Gravity acting on a 10 kg mass produces a force of F_g = á 0,-98 ñ newtons. If the mass is suspended from 2 wires which both form 30^° angles with the horizontal, then what forces of tension are required in order for the mass to hang motionless over time?

30. Repeat exercise 29 given angles of 45^° instead of 30^°.

31. A 10 kg block sits on a board inclined to an angle of 30^° with the horizontal.
What are F_inc (the force along the incline) and F_n (the force normal)? 

32. Repeat exercise 31 given an incline of 10^° instead of 30^°.            

33. Given P₁( 1,2,1) and v = á 3,2,1 ñ , find the point P₂ for which

P1P2   = v

34. Given P₂( 5,5,5) and v = á 5,-2,1 ñ , find the point P₁ for which

P1P2   = v

35. In this exercise, we explore the difference of 2 vectors.

1. Let v be the vector with initial point P₁(3,5) and final point P₂( 5,1) . What is v? 
2. Let u be the vector with initial point P₁(3,5) and final point P₃( 6,7) . What is u?
3. Compute u-v and show that it is the same as the vector from P₂( 3,5) to P₃( 6,7) .
4. Sketch a graph of the system.

36.  Let's prove that in general, the difference of 2 two dimensional vectors is the cross-diagonal of the parallelogram they form. Because vectors are translation invariant, we need only do so for vectors with initial points at the origin.

Let v be the vector from P₁( 0,0) to P₂( a,b) , and let u be the vector from P₁( 0,0) to P₃( c,d) . What is the vector from P₂( a,b) to P₃( c,d) ? How is it related to u-v?

37. In this exercise, we explore the sum of 2 vectors.

1. Let v be the vector with initial point P₁(0,0) and final point P₂( 5,1) . What is v?  
2. Let u be the vector with initial point P₁(0,0) and final point P₃( 3,5) . What is u?
3. Compute u+v, and then sketch a graph of the system, including the parallelogram formed by u and v.

38.  Let's prove that in general, the sum of 2 two dimensional vectors is the main diagonal of the parallelogram they form. Because vectors are translation invariant, we need only do so for vectors with initial points at the origin.

1. Explain why the line through ( 0,0) and (a,b) is parallel to the line through ( c,d) and ( a+c,b+d) .
2. Explain why the line through ( 0,0) and (c,d) is parallel to the line through ( a,b) and ( a+c,b+d) . Why does this imply that ( a+c,b+d) is the fourth vertex of the parallelogram with vertices ( 0,0) , ( a,b) , and ( c,d) .
3. Let v be the vector from P₁( 0,0) to P₂( a,b) , and let u be the vector from P₁( 0,0) to P₃( c,d) . Find u+v and relate it to the diagram above? 

39. Let u = áu₁,u₂,u₃ ñ , v = áv₁,v₂,v₃ ñ , and w = áw₁,w₂,w₃ ñ . Prove properties 1, 3, and 4 of theorem 1.1

40. Let k,m be scalars and let u = áu₁,u₂,u₃ ñ , v = áv₁,v₂,v₃ ñ . Prove properties 5, 6, and 7 of theorem 1.1

41. A 10 kg block is at rest on a frictionless incline at 30^° to the horizontal, held in place by a rope attached to a mass hanging off the end of the incline.

What is the mass of the block at the other end of the rope?

42. Class Discussion:  A block at rest on an incline experiences a force of static friction F_static with a maximum magnitude of

||Fstatic|| = ms||Fn||

(3)

where F_n is the force normal (perpendicular) to the incline pushing üp" against the block and m_s is the coefficient of static friction of the incline's surface.

Use the diagram to explain how you could estimate m_s experimentally by slowly lifting the end of the incline (i.e. varying a) until the block begins to slide

43. A vector space is a set V which is closed under an addition operation and a scalar multiplication operation which jointly satisfy theorem 1.1. Show that the set of functions

V = { f  :  f  is  continuous  on  [ 0,1]   }

is a vector space by showing that it is closed under addition, closed under multiplication by a number, and satisfies the properties in theorem 1.1.

44. A vector space is a set V which is closed under an addition operation and a scalar multiplication operation which jointly satisfy theorem 1.1. Show that the set of functions

V = { f( x) = a+bx+cx2,  where  a,b,c  are  numbers  }

is a vector space by showing that it is closed under addition, closed under multiplication by a number, and satisfies the properties in theorem 1.1.