0.2 Scalar Product

1
0.2 Scalar Product ProjectionsSt Bk
Readings in App A Stewart Ch 9_________________
__________________________________________________
__________________________________________________
____________

• What Why?
• --------------------------
• Scalar or Dot Product is a single number that
  holds information about the angle between 2
  vectors.
• We can use it to find that angle or to calculate
  lengths
• And to test for perpendicularity very
  important!
• And to calculate Projections Components. These
  tell us how far a given vector extends in some
  direction of interest!

2

• Defn of Dot (or Scalar) Product
• (a1, a2) . (b1,b2) a1b1 a2b2
  NB
• eg ( 3, -1) . ( 2, 4)
  6 - 4 2.
• The dot product of a vector with itself gives
  the sq of its length
• (a, b) . (a. b) a2 b2
  .
• ie u . u
  u2 NB
• Know the Rules for Dot Product Study Book
  p12.
• Defn of the angle between two vectors
• it is the smallest non-negative (ie
  unsigned) angle.
• Using the Cosine Rule
• Th 2, App A, p 166, proves u . v u
  v cos t NB!
• Hence dot product can be used to find angle t
  cos t u.v

3

• In u . v u v cos t ,
• the lengths u v are always ve.
• Hence the SIGN of u . v is determined by the
  factor cos t .
• Now for acute angles, cos t is ve
• for obtuse angles, cos t is -ve.
• Hence the sign of their dot product of tells us
  about the size of the angle between two vectors
  t
• they pull together if and only if their dot
  product is ve.
• they pull apart if and only if their dot
  product is -ve.
• t

4

• Tests for Parallel Perpendicular vectors
• Parallel vectors have angle 0 or ? between
  them
• They are scalar multiples of each other (App A,
  Th3, p 167).
• Example (3, -1) (-6, 2) are
  parallel
• see that (-6, 2) - 2 (3, -1) .
• Orthogonal (ie perpendicular) vectors make a
  right angle.
• Substituting cos t 0 for t ?/2 into
  cos t u . v / uv
• gives the dot product test for non-zero
  vectors u v
• u v are perpendicular if only
  if u . v 0.

5

• Scalar Component of u in the direction of v
• u
• t v
• this length ???
• How far does u project in the direction of v?
• Using trig, this scalar component is u
  cos t NB
• But since u . v u v cos t
• another way to calculate scalar component is
  u .v / v .
• Eg the scalar projection of (3,1) on ( 3, 4)
  is (94) / 5 .
• Scalar projection/component is a signed
  distance
• because cos t is ve if t is acute,
• - ve if t is obtuse.

6

• Vector Projection of u on v
• To express the scalar component or projection as
  a vector quantity, we point it in the direction
  of unit vector v
• v
• Projv u u . v v u
  . v v
• v v v 2
• We can then decompose (or resolve) u into the
• sum of two orthogonal vectors u
  
• u-p
• p
• One is the projection p , which can be found by
  this formula. The other is then simply u - p ,
  found by subtraction.

7
Examples Exercises
8
Homework Read Study Book Section 0.2

• Without a calculator, give exact values for the
  sine, cos tan of 0 , ?/2 , ? , ?/6 ,
  ?/4 , ?/3 , 3? /4 , -? /4 .
• The 30/60/90 degree 45/45/90 triangles are
  a big help
• 1 sqrt2 2 2
• sqrt 3
• 1 1 1
• Appendix A Problems 3.2 Master 1-16, 21-26,
  39.
• Write full solutions to Q 2, 9, 10, 12, 14, 16,
  34, 39.
• Study Book Th 1 p 12 try to prove the rules.

9

• Objectives
• Be able to
• use dot product to find angles
• to determine the relative direction of 2
  vectors
• spot parallel vectors - as multiples of each
  other
• use dot product to test for perpendicularity
• prove use the rules for dot product
• find the scalar component of one vector on
  another
• find the vector projection of one on another
• decompose a vector into the sum of two that are
  
• mutually orthogonal