Advanced Precalculus Notes 8.5 The
Dot Product
The dot product of two vectors is a scalar:
v  w  a1a2  b1b2
If v = 2i – 3j
and w = 5i + 3j
find:
a) v ∙ w
b) w ∙ v
c) v ∙ v
d) w ∙ w
e) ||v||
f) ||w||
Properties of dot products:
Commutative Property: u ∙ v = v ∙ u
Distributive Property: u ∙ (v + w) = u ∙ v + u ∙ w
v ∙ v = ||v||2
Angle between Vectors:
0∙v=0
cos 
uv
|| u ||  || v ||
Find the angle between u = 4i -3j and v = 2i + 5j
A Boeing 737 aircraft maintains a constant airspeed
of 500 mph due South. The velocity of the jet
stream is 80 mph in a northeasterly direction. Find
the actual speed and direction of the aircraft
relative to the ground.
Parallel Vectors: if
parallel.
uv
|| u ||  || v ||
= 1 or -1, the vectors are
Orthogonal Vectors: If the dot product is zero, the vectors
are orthogonal (perpendicular).
v = 3i – j
w = 6i – 2j
u = 2i – j
z = 3i + 6j
a) Are vectors v and w parallel, orthogonal of neither?
b) Are vectors u and z parallel, orthogonal or neither?
vw
w
Vector projection of v onto w: v1 
2
|| w ||
Find the vector projection of v = i +3j
v2  v  v1
onto w = i + j
Work: work W done by a constant force F is:
W = (magnitude of force)(distance) = F AB  F  AB
An object is pulled with a force of 50 pounds. How
much work is done in moving the object 100 feet if
the handle makes an angle or 30º with the ground?
Assignment:
page 626:
1 – 7, 12, 16, 17, 19, 21,
25, 27, 29, 32, 35