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APPLICATIONS OF TRIGONOMETRY
CHAPTER SIX
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VECTORS IN THE PLANE
SECTION 6.1
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MAGNITUDE & DIRECTION
• Temperature• Height• Area• Volume
• SINGLE REAL NUMBER INDICATING SIZE
• Force• Velocity• Acceleration
• MAGNITUDE AND DIRECTION
NEED TWO NUMBERS
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<a,b>Position Vector of (a,b)
Length represents magnitude, and the direction in which it points represents direction
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Two-Dimensional VectorA two-dimensional vector v is an ordered pair of real numbers denoted in component form as <a,b>. The numbers a and b are the components of vector v.
The standard representation of the vector is the arrow from the origin to the point (a,b).
The magnitude of v is the length of the arrow and the direction of v is the direction in which the arrow is pointing.
The vector 0 is called the zero vector – zero length and no direction
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Any two arrows with the same length and pointing in the same direction represent the same vector.
Equivalent vectors
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Head Minus Tail Rule for VectorsIf an arrow has initial point (x1,y1) and terminal point (x2,y2), it represents the vector <x2-x1, y2-y1>
Example 11. An arrow has initial point (2,3) and terminal point (7,5).
What vector does it represent?2. An arrow has initial point (3,5) and represents the vector
<-3, 6>. What is the terminal point?3. If P is the point (4,-3) and PQ represents <2, -4>, find Q.4. If Q is the point (4,-3) and PQ represents <2,-4>, find P.
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Magnitude of a Vector, v ¿𝑣∨¿√∆ 𝑥2+∆ 𝑦2
If v = <a,b>, then |v|=
Example 2 Find the magnitude of the vector v represented by , where P = (-2, 3) and Q = (-7,4).
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Vector Operations• When we work with vectors and numbers at the same time we
refer to the numbers as scalars.
• The two most common and basic operations are vector addition and scalar multiplication.
• Vector Addition• Let u = < and v = <, the sum (or resultant) of the vectors is
u + v = <>• The product of the scalar k and the vector u is
ku = k < = <
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Vector Operation ExamplesExample 3: Let u = <-2,5> and v = <5,3>. Find the component form of the following vectors:a. u + v b. 4u c. 3u + (-1)v
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Unit Vector
A vector u with length |u|=1.
u =
Example 4: Find a unit vector in the direction of v = <-4,6> and verify it has a length of 1.
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Standard Unit Vectors
i = <1,0> j = <0,1>
Any vector v can be written as an expression in terms of the standard unit vectors.
v = <a,b>= <a,0> + <0,b> = a <1,0> + b <0,1> = ai + bj
The scalars a and b are the horizontal and vertical components of the vector v.
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Direction AnglesUsing trigonometry we can resolve the vector.
Find the direction angle. That is, the angle that v makes with the x-axis.
Vertical & Horizontal component
If v has direction angle θ, the components of v can be computed using the formula
v = <|v|cosθ, |v|sinθ >
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Ex. 5: Find Components of a VectorFind the components of a vector v with direction angle 135 degrees and magnitude 10.
Ex. 6: Find Direction Angle of VectorFind the magnitude and direction angle of each vector:
(a) u = <-4,6> (b) v = <5,7>
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HOMEWORK: p. 464: 3-27 multiples of 3,
29, 34, 37, 42, 43, 49
p. 472: 1-19 odd, 21-24
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Dot Product of Vectors
SECTION 6.2
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Vector Multiplication
Cross Product• Results in a vector
perpendicular to the plane of the two vectors being multiplied
• Takes us into a third dimension
• Outside the scope of this course
Dot Product• Results in a scalar• Also known as the “inner
product”
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Dot Product
The dot product or inner product of u = < and v = <u · v =
Example: Find each dot product.a. <4,5 ·<2, 3> b. <-1,3 ·<2i, 3j>
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Properties of the Dot Product
Let u, v, and w be vectors and let c be a scalar.
1. u · v = v · u
2. u · u =
3. 0 · u = 0
4. u · (v + w) = u · v + u · w
5. (cu) · v = u · (cv) = c(u · v)
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Angle Between Two Vectors
If θ is the angle between the nonzero vectors u and v, then
cos θ =
and θ = )
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Example: Finding the Angle Between Two VectorsUse an algebraic method to find the angle between the vectors u and v.
a. u = <4, 1>, v = <-3, 2>
b. u = <3, 5>, v = <-2, -4>
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Orthogonal Vectors
The vectors u and v are orthogonal if and only if u · v = 0.
Note: Orthogonal means basically the same thing as perpendicular.
Example: Prove that the vectors u = <3, 6> and v = <-12, 6> are orthogonal
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Projection of a Vector
If u and v are nonzero vectors, the projection of u onto v is
)v