# 0.2 Scalar Product Projections

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```					     0.2 Scalar Product & Projections
St Bk + Readings in App A & Stewart Ch 9
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What & Why?
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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!
1
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       = |u|2       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
NB!                       | u| |v |
2
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
3
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 / |u||v|
gives the dot product test for non-zero vectors u & v:
u & v are perpendicular if & only if u . v = 0.
4
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     (9+4) / 5 .
Scalar projection/component is a “signed” distance
because cos t      is +ve if t is acute,
- ve if t is obtuse.
5
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.
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Examples & Exercises:

7
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.
8
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

9

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