Dot Product

(requires JavaScript)

  1. Which of the following expression are meaningful? Which are meaningless? Explain.
    1. abc
    2. abc
    3. abc
    4. ab+c
    5. ab+c
    6. ab+c
    b, c, and d are meaningful.
  2. Find the dot product of two vectors if their lengths are 6 and 13 and the angle between them is π4 .
  3. Find ab if a=4114 , b=638 .
  4. Find ab if a=4j3k and b=2i+4j+6k .
  5. If u is a unit vector, find uv and uw .
    unit vector square

    From the figure, u=w=1 , and v=12 . The angle between u and w is π/2 , and the angle between u and v is π/4 . So we can write




  6. Find the angle between vectors a=j+k and b=i+2j3k .
  7. Determine whether given vectors are orthogonal, parallel, or neither.
    1. a=537 , b=682 .
    2. a=46 , b=32 .
    3. a=i+2j+5k , b=3i+4jk .
    4. a=2i+6j4k , b=3i9j+6k .
    Neither, orthogonal, orthogonal, parallel.
  8. Find two unit vectors that make an angle of 60° with v=34 .
  9. Find the scalar and vector projections of b onto a , where a=i+j+k and b=ij+k .

    compab=13 ,

    projab=13111 .

  10. Suppose that a and b are non-zero vectors. Under what circumstances is
    1. compab=compba
    2. projab=projba
  11. An intelligent robot exerts a horizontal force of 25 lb on a crate as it pushes it up a ramp that is 10 feet long and inclined at an angle of 20° above the horizontal. Find the work done on the box.
    250cos20°235 ft-lb.
  12. Find the angle between a diagonal of a cube and one of its edges.