VECTOR ALGEBRA
MATH23
MULTIVARIABLE CALCULUS
GENERAL OBJECTIVE
At the end of the lesson the students are expected to:
• Perform dot product and cross product
• Apply the concepts of dot product to get the angle between
two vectors
• Apply cross products in various applications
• Evaluate scalar triple product
The Dot (Scalar) Product
Definition:
If u = <u1, u2 > and v = <v 1, v2> are vectors in
2-space, then the dot product of u and v, written
as u.v, is defined as
uv = u1 v1 + u2 v2
Similarly, if u = <u1 , u2 , u3 > and v = <v1 , v2, v3
> are vectors in 3-space, then their dot product is
defined as
u  v = u1 v1 + u2 v2 + u3 v3
The Dot (Scalar) Product
Angle Between Two Vectors
Let u and v be nonzero vectors in 2-space or 3space, and if θ is the angle between them, then
u v
cos 
u v
Note:
1) If uv > 0, θ is an acute angle.
2) If u  v < 0 , θ is an obtuse angle.
3) If u  v = 0, θ is a right angle.
.
Direction Angles
In an xy-coordinate system, the direction of a
nonzero vector v is determined by the angles α and β
between v and the unit vectors i and j, and in an xyzcoordinate system, the direction is determined by the
angles α, β, and γ between v and the unit vectors i, j, and k.
In both 2- space and 3-space, the angles between a nonzero
vector v and the vectors i , j, and k are called the direction
angles of v, and the cosines of those angles are called the
direction cosines of v.
The Dot (Scalar) Product
Direction Angles
Using angle between two vectors, ,,and 
are obtained as follows:
vi
cos 
,
v
v1
  cos
v
 v1 , v2 , v3    1,0,0 

v
v1

v
-1
The Dot (Scalar) Product
cos  
v j
,
v
  cos-1

 v1 , v2 , v3    0,1,0 
v

v2
v
vk
cos 
,
v
v3
  cos
v
 v1 , v2 , v3    0,0,1 

v
v3

v
v2
v
-1
Exercise Set 11.3
1. Find the dot product of the vectors and the cosine of
the angle between them.
b) u = <-7, 3>, v = <0, 1>
c) u = 1- 3j + 7k, v = 8i – 2j – 2k
2. a) Find u.v if ǁuǁ = 1, ǁvǁ = 2, the angle between u and v
is π/6.
3. Determine whether u and v make an acute angle, an
obtuse angle, or are orthogonal.
b) u = 6i +j + 3k, v = 4i – 6k
7. b) Use vectors to find the interior angles of the triangle
with vertices (-1, 0), (2, -1), and (1, 4).
Exercise Set 11.3
13. Find r so that the vector from the point
A (1, -1, 3) to the point B (3, 0, 5) is orthogonal
to the vector from A to the point P (r, r, r).
16. Find the direction cosines and direction
angles of a) v = 3i – 2j – 6. b) v = 3i – 4k.
The Cross (Vector) Product
Let u=<u1, u2, u3>, and v=<v1, v2, v3> be vectors
in 3-space. The cross product of u and v,
denoted by uxv, is defined as
i
j
k
uxv  u1 u2 u3
v1 v 2 v 3
The Cross (Vector) Product
Algebraic Properties of the Cross Product
Theorem: If u, v, and w are any vectors in 3space and k is any scalar, then:
a) u x v = - (v x u)
b) u x (v + w) = (u x v) + (u x w)
c) (u + v) x w = ( u x w) + (v x w)
d) k(u x v) = (ku) x v = u x (kv)
e) u x 0 = 0 x u = 0
f) u x u = 0
The Cross (Vector) Product
Geometric Properties of the Cross Product
Theorem:
1. If u and v are vectors in 3-space, then:
a) u. (u x v) = 0
(u x v is orthogonal to u)
b) v. (u x v) = 0
(u x v is orthogonal to v)
2. Let u and v be nonzero vectors in 3-space, and let θ be the angle
between these vectors when they are positioned so their initial points
coincide.
a) ǁu x vǁ = ǁuǁ ǁvǁ sinθ
b) The area A of a parallelogram that has u and v as adjacent sides is
A=ǁuxvǁ
c) u x v = 0 if and only if u and v are parallel vectors, that is, if and
only if they are scalar multiples of one another.
Exercise Set 11.4
4. Find u x v and check that it is orthogonal to both
u = 3i +2j - k and v = - 1i – 3j + k.
7. Let u = <2, -1, 3>, v =<0, 1, 7> and w = <1, 4, 5>.
Find
a) u x ( v x w) b) (u x v) x (v x w)
10. Find two unit vectors that are orthogonal to
both
u = -7i + 3j + k, v = 2i + 4k.
18. Find the area of the parallelogram that has u =
2i + 3j and v = -i + 2j – 2k as adjacent sides.
20. Find the area of the triangle with vertices P(2,
0, -3), Q(1, 4, 5), R(7, 2, 9).
The Scalar Triple Products
• The scalar triple product is defined as the dot product of one
of the vectors with the cross product to the third vector.
Geometric interpretation
• Geometrically, the scalar triple product is the (signed) volume
of the parallelepiped defined by the three vectors given.
PROPERTIES OF SCALAR TRIPLE PRODUCT
• The scalar triple product can be evaluated numerically using
any one of the following equivalent equations:
• Switching the two vectors in the cross product negates the
triple product,
• The parentheses may be omitted without causing ambiguity,
since the dot product cannot be evaluated first. If it were, it
would leave the cross product of a scalar and a vector, which
is not defined.
Properties of Scalar Triple Product
• The scalar triple product can also be understood as the determinant of the
3 × 3 matrix having the three vectors as its rows or columns (the
determinant of a transposed matrix is the same as the original); this
quantity is invariant under coordinate rotation.
• Note that if the scalar triple product is equal to zero, then the three
vectors a, b, and c are coplanar, since the "parallelepiped" defined by
them would be flat and have no volume
Exercise Set 11.4
24. Find u . (v x w) where u = i, v = i + j, w = i + j + k.
26. Find the volume of the parallelepiped that has
u = 3i + j + 2k, v = 4i + 5j + k and, w = i + 2j + 4k as
adjacent sides.
TEXTBOOKS
SUGGESTED READINGS
Anton, Howard; Bivens Irl and Davis Stephen Calculus,
Early Transcendentals, Chapter 7 pages 547 to 555
Peterson, Thurman S Calculus With Analytic Geometry,
Chapter 14 pages 289 to 292