January 06, 2015
Section 6.4 Vectors and Dot Products
Objectives: Find the dot product of two vectors, use properties of the
dot product, and find the angle between two vectors.
The Dot Product of Two Vectors
The dot product of u = u , u
1
2
and v = v , v
1
2
is given by
u v=u v +u v.
1
1
2
2
*Notice this results in a scalar, not a vector, and can be positive, 0, or negative.
Properties of the Dot Product
Let u, v, and w be vectors in the plane or in space and let c be a
scalar.
1. u v = v u
2. 0 v = 0
3. u (v + w) = u v + u w
4. v v = ||v||
5. c(u v) = cu v = u cv
2
Ex1: Let u = 2, 6 , v = -1, 5 , and w = -3, 1 . Find the following.
a) u v
a) u v = 2, 6 -1, 5 = 2(-1)
b) u (v w)
b) u (v w) = 2, 6 8 = 16, 4
c) u (v + w)
Ex2: The dot product of u with itself is 3. What is the magnitude of u?
b) u (v + w) = u v + u w =
u u = 3, llull2 = 3, llull = √3
The Angle Between Two Vectors
The angle between two nonzero vectors is the angle θ, 0 ≤ θ ≤ 2π,
between their respective standard position vectors.
v-u
u v
To find...
cos θ =
llull llvll
u
θ
Origin
Ex3: Find the angle between u = -2, 3 and v = 1, -5 .
v
January 06, 2015
Rewriting the angle formula produces an alternative way to calculate
the dot product, u v = ||u|| ||v|| cos θ.
Note u v and cos θ will always have the same sign.(5 possible orientations, p. 424)
Orthogonal Vectors
Since cos 90° = 0, we can say that two vectors u and v are orthogonal
(perpendicular) if u v = 0. Note that this implies that the zero vector
is orthogonal to every vector.
Ex3: Which vectors, if any, are orthogonal?
a) u = 1, -4 and v = 6, 2
a) u v = 1(6) + -4
b) u = -4, 2 and v = 1, 2
b) u v = -4(1) + 2
Finding Vector components
Do the reverse of adding two vectors to find their resultant
a vector into the sum of vector components. How?
decompose
Definition of Vector Components
Let u and v be nonzero vectors such that u = w + w , where w and w are
1
2
1
2
orthogonal and w is parallel to (or a scalar multiple of) v. The vectors w and w
1
1
are called vector components of u. The vector w is the projection of u onto v
1
and is denoted by w = proj u.
1
v
w2
The vector w is given by w = u - w .
2
2
1
u
u
θ
v
w1
w1
w2
θ
v
Let u and v be nonzero vectors. The projection of u onto v is given by...
proj u =
v
( )
u v
||v||2
v.
Ex4: Find the projection of u = 5, 2 onto v = 3, -1 . Then write u as
the sum of two orthogonal vectors, one of which is proj u.
v
2
January 06, 2015
Ex5: A 500-pound piano sits on a ramp that is inclined at 45°. What
force is required to keep the piano from rolling down the ramp?
Force required to
keep piano from
rolling down ramp.
v
w =?
1
45
We need the projection o
w
2
Force against ramp
F = -500j
Force of gravity
(weight of piano)
Work
Work W done by a constant force F as its point of application moves
along the vector is given by of the following:
W=
Projection form
= (cos θ)llFll llPQll
W=F
http://www.ajdesigner.com/phpwork/work_equation.php
http://engineering-students.com/formula_solvers.php
Dot product form
Ex6: Frank pushes a broom with a constant force of 40 pounds and holds
the handle of the broom at an angle of 30°. How much work is done
pushing the broom 30 feet?
W = F 30i = 40(cos 30°i + sin 30°j) 30i = 1200cos 30° ≈ 1039.23 footpounds.
p. 429 (3-51x3, 53, 56, 58, 60)
So, the force required to k
is 250 √2 ≈ 353.6 pounds
Dot product of F and v