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7
Applications of
Trigonometry
and Vectors
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7.5 Algebraically Defined Vectors
and the Dot Product
Algebraic Interpretation of Vectors ▪ Operations with Vectors ▪
The Dot Product and the Angle between Vectors
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Algebraic Interpretation of
Vectors
 A vector with its initial point at the origin is called
a position vector.
 A position vector u with its endpoint at the point
(a, b) is written
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Algebraic Interpretation of
Vectors
 The numbers a and b are the horizontal
component and vertical component,
respectively, of vector u.
 The positive angle
between the x-axis and a
position vector is the
direction angle for the
vector.
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Magnitude and Direction Angle
of a Vector a, b
The magnitude (length) of a vector u = a, b
is given by
The direction angle θ satisfies
where a ≠ 0.
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Example 1
FINDING MAGNITUDE AND DIRECTION
ANGLE
Find the magnitude and direction angle for u = 3, –2.
Magnitude:
Direction angle:
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Example 1
FINDING MAGNITUDE AND DIRECTION
ANGLE (continued)
Graphing calculator solution:
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Horizontal and Vertical Components
The horizontal and vertical components,
respectively, of a vector u having magnitude
|u| and direction angle θ are given by
or
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Example 2
FINDING HORIZONTAL AND VERTICAL
COMPONENTS
Vector w has magnitude 25.0 and direction angle
41.7°. Find the horizontal and vertical components.
a  w cos 
b  w sin
 25.0 cos 41.7
 25.0 sin 41.7
 18.7
 16.6
Horizontal component: 18.7
Vertical component: 16.6
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Example 2
FINDING HORIZONTAL AND VERTICAL
COMPONENTS
Graphing calculator solution:
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Example 3
WRITING VECTORS IN THE FORM a, b
Write each vector in the figure in
the form a, b.
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Vector Operations
Let a, b, c, d, and k represent real numbers.
If u  a1, a2 , then  u  a1, a2 .
a, b  c, d  a, b    c, d

a  c, b  d
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Example 4
PERFORMING VECTOR OPERATIONS
Let u = –2, 1 and v = 4, 3. Find and illustrate the
following.
(a) u + v = –2, 1 + 4, 3 = –2 + 4, 1 + 3 = 2, 4
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Example 4
PERFORMING VECTOR OPERATIONS
Let u = –2, 1 and v = 4, 3. Find and illustrate the
following.
(b) –2u = –2 ∙ –2, 1 = –2(–2), –2(1) = 4, –2
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Example 4
PERFORMING VECTOR OPERATIONS
Let u = –2, 1 and v = 4, 3. Find and illustrate the
following.
(c) 3u – 2v = 3 ∙ –2, 1 – 2 ∙ 4, 3
= –6, 3 –8, 6
= –6 – 8, 3 – 6
= –14, –3
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Unit Vectors
 A unit vector is a vector that has magnitude 1.
i = 1, 0
j = 0, 1
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Unit Vectors
Any vector a, b can be expressed in the form
ai + bj using the unit vectors i and j.
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i, j Form for Vectors
If v = a, b, then
v = ai + bj, where i = 1, 0 and j = 0, 1.
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Dot Product
The dot product of the two vectors
u = a, b and v = c, d is denoted
u ∙ v, read “u dot v,” and is given by the
following.
u ∙ v = ac + bd
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Example 5
FINDING DOT PRODUCTS
Find each dot product.
(a) 2, 3 ∙ 4, –1 = 2(4) + 3(–1) = 5
(b) 6, 4 ∙ –2, 3 = 6(–2) + 4(3) = 0
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Properties of the Dot Product
For all vectors u, v, and w and real
numbers k, the following hold.
(a) u ∙ v = v ∙ u
(b) u ∙ (v + w) = u ∙ v + u ∙ w
(c) (u + v) ∙ w = u ∙ w + v ∙ w
(d) (ku) ∙ v = k(u ∙ v) = u ∙ (kv)
(e) 0 ∙ u = 0
(f) u ∙ u = |u|2
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Geometric Interpretation of
the Dot Product
If θ is the angle between the two nonzero
vectors u and v, where 0° ≤ θ ≤ 180°, then
the following holds.
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Example 6(a) FINDING THE ANGLE BETWEEN TWO
VECTORS
Find the angle θ between the two vectors u = 3, 4
and v = 2, 1.
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Example 6(b) FINDING THE ANGLE BETWEEN TWO
VECTORS
Find the angle θ between the two vectors u = 2, –6
and v = 6, 2.
2, 6
u v
cos  

u v
2, 6

6, 2
6,2
2(6)  ( 6)(2)
4  36 36  4
0

0
40
  cos1 0  90
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Dot Products
For angles θ between 0° and 180°, cos θ is positive, 0,
or negative when θ is less than, equal to, or greater
than 90°, respectively.
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Note
If u ∙ v = 0 for two nonzero vectors u
and v, then cos θ = 0 and θ = 90°.
Thus, u and v are perpendicular or
orthogonal vectors.
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