0.2 Scalar Product & Projections
St 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!
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 |
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.
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.
Scalar Component of u in the direction of 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.
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
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
One is the projection p , which can be found by this formula.
The other is then simply u - p , found by subtraction.
Examples & Exercises:
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
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.
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