APPLICATIONS OF TRIGONOMETRY

CHAPTER SIX

VECTORS IN THE PLANE

SECTION 6.1

MAGNITUDE & DIRECTION

• Temperature• Height• Area• Volume

• SINGLE REAL NUMBER INDICATING SIZE

• Force• Velocity• Acceleration

• MAGNITUDE AND DIRECTION

NEED TWO NUMBERS

<a,b>Position Vector of (a,b)

Length represents magnitude, and the direction in which it points represents direction

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

Any two arrows with the same length and pointing in the same direction represent the same vector.

Equivalent vectors

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.

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).

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 < = <

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

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.

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.

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 >

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>

HOMEWORK: p. 464: 3-27 multiples of 3,

29, 34, 37, 42, 43, 49

p. 472: 1-19 odd, 21-24

Dot Product of Vectors

SECTION 6.2

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”

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>

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)

Angle Between Two Vectors

If θ is the angle between the nonzero vectors u and v, then

cos θ =

and θ = )

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>

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

Projection of a Vector

If u and v are nonzero vectors, the projection of u onto v is

)v