Engineering Math

3.8 Exercises

1. Show that if a × u = 0 for all unit vectors u, then a = 0.
2. If you only assume 3.30 holds for u = i,j,k, show that this implies 3.30 holds for all unit vectors u.
3. Let m₁ = 5,m₂ = 1, and m₃ = 4 where the masses are in kilograms and the distance is in meters. Suppose m₁ is located at 2i − 3j + k, m₂ is located at i − 3j + 6k and m₃ is located at 2i + j + 3k. Find the center of mass of these three masses.
4. Let m₁ = 2,m₂ = 3, and m₃ = 1 where the masses are in kilograms and the distance is in meters. Suppose m₁ is located at 2i − j + k, m₂ is located at i − 2j + k and m₃ is located at 4i + j + 3k. Find the center of mass of these three masses.
5. Find the angular velocity vector of a rigid body which rotates counter clockwise about the vector i−2j + k at 40 revolutions per minute. Assume distance is measured in meters.
6. Let
   {u1,u2,u3}
   be a right handed system with u₃ pointing in the direction of i−2j + k and u₁ and u₂ being fixed with the body which is rotating at 40 revolutions per minute. Assuming all distances are in meters, find the constant speed of the point of the body located at 3u₁+u₂−u₃ in meters per minute.
7. Find the area of the triangle determined by the three points
   (1,2,3)
   ,
   (4,2,0)
   and
   (− 3,2,1)
   .
8. Find the area of the triangle determined by the three points
   (1,0,3)
   ,
   (4,1,0)
   and
   (− 3,1,1)
   .
9. Find the area of the triangle determined by the three points
   (1,2,3)
   ,
   (2,3,4)
   and
   (0,1,2)
   . Did something interesting happen here? What does it mean geometrically?
10. Find the area of the parallelogram determined by the vectors
    (1,2,3)
    and
    (3,− 2,1)
    .
11. Find the area of the parallelogram determined by the vectors
    (1,0,3)
    and
    (4,− 2,1)
    .
12. Find the area of the parallelogram determined by the vectors
    (1,− 2,2)
    and
    (3,1,1)
    .
13. Find the volume of the parallelepiped determined by the vectors i−7j−5k,i−2j−6k,3i+2j+3k.
14. Find the volume of the parallelepiped determined by the vectors i+j−5k,i+5j−6k,3i+j+3k.
15. Find the volume of the parallelepiped determined by the vectors i+6j+5k,i+5j−6k,3i+j+k.
16. Suppose a,b, and c are three vectors whose components are all integers. Can you conclude the volume of the parallelepiped determined from these three vectors will always be an integer?
17. What does it mean geometrically if the box product of three vectors gives zero?
18. Find the equation of the plane through the three points
    (1,2,3)
    ,
    (2,− 3,1)
    ,
    (1,1,7)
    .
19. It is desired to find an equation of a plane containing the two vectors a and b and the point 0. Using Problem 17, show an equation for this plane is

    | | || x y z || ||a1 a2 a3 ||= 0 ||b b b || 1 2 3

    That is, the set of all

    (x,y,z)
    such that

    | | | | | | || a2 a3 || || a1 a3 || ||a1 a2 || x|| b2 b3 ||− y|| b1 b3 ||+ z|| b1 b2 || = 0

20. Using the notion of the box product yielding either plus or minus the volume of the parallelepiped determined by the given three vectors, show that

    (a × b)⋅c = a⋅(b× c)

    In other words, the dot and the cross can be switched as long as the order of the vectors remains the same. Hint: There are two ways to do this, by the coordinate description of the dot and cross product and by geometric reasoning.

21. Is a×
    (b× c)
    =
    (a× b)
    × c? What is the meaning of a × b × c? Explain. Hint: Try
    (i× j)
    ×j.
22. Verify directly that the coordinate description of the cross product a × b has the property that it is perpendicular to both a and b. Then show by direct computation that this coordinate description satisfies

    ( ) |a ×b |2 = |a|2|b |2 − (a ⋅b)2 = |a|2|b|2 1− cos2 (θ)

    where θ is the angle included between the two vectors. Explain why

    |a × b|
    has the correct magnitude. All that is missing is the material about the right hand rule. Verify directly from the coordinate description of the cross product that the right thing happens with regards to the vectors i,j,k. Next verify that the distributive law holds for the coordinate description of the cross product. This gives another way to approach the cross product. First define it in terms of coordinates and then get the geometric properties from this.

23. Discover a vector identity for u×
    (v× w )
    .
24. Discover a vector identity for
    (u × v)
    ⋅
    (z× w )
    .
25. Discover a vector identity for
    (u × v)
    ×
    (z× w )
    in terms of box products.
26. Simplify
    (u× v)
    ⋅
    (v × w )
    ×
    (w × z)
    .
27. Simplify
    |u × v|
    ² +
    (u ⋅v)
    ² −
    |u|
    ²
    |v|
    ².
28. Prove that ε_ijkε_ijr = 2δ_kr.
29. If A is a 3 × 3 matrix such that A =
    ( ) u v w
    where these are the columns of the matrix A. Show that det
    (A )
    = ε_ijku_iv_jw_k.
30. If A is a 3 × 3 matrix, show ε_rps det
    (A )
    = ε_ijkA_riA_pjA_sk.
31. Suppose A is a 3 × 3 matrix and det
    (A)
    ≠0. Show using 30 and 28 that

    (A−1) = ---1---εrpsεijkApjAri. ks 2det(A)

32. When you have a rotating rigid body with angular velocity vector Ω then the velocity, u^′ is given by u^′ = Ω × u. It turns out that all the usual calculus rules such as the product rule hold. Also, u^′′ is the acceleration. Show using the product rule that for Ω a constant vector

    u′′ = Ω× (Ω × u).

    It turns out this is the centripetal acceleration. Note how it involves cross products.

33. Find the planes which go through the following collections of three points. In case the plane is not well defined, explain why.
    1. (1,2,0)
       ,
       (2,− 1,1)
       ,
       (3,1,1)
    2. (3,1,0)
       ,
       (2,1,1)
       ,
       (− 3,1,− 1)
    3. (2,1,1)
       ,
       (− 2,3,1)
       ,
       (0,4,2)
    4. (1,0,1)
       ,
       (2,0,1)
       ,
       (0,1,1)
34. A point is given along with a line. Find the equation for the plane which contains the line as well as the point.
    1. (1,2,1)
       ,
       (1,− 1,1)
       + t
       (1,0,1)
    2. (2,1,− 1)
       ,
       (1,1,1)
       + t
       (2,− 1,1)
    3. (− 1,2,3)
       ,
       (− 1,1,1)
       + t
       (2,1,1)
    4. (2,0,1)
       ,
       (2,1,1)
       + t
       (− 1,1,1)