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 Vector A 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 Vectors If an arrow has initial point (x1,y1) and terminal point (x2,y2), it represents the vector <x2-x1, y2-y1> Example 1 An arrow has initial point (2,3) and terminal point (7,5). What vector does it represent? An arrow has initial point (3,5) and represents the vector <-3, 6>. What is the terminal point? If P is the point (4,-3) and PQ represents <2, -4>, find Q. If Q is the point (4,-3) and PQ represents <2,-4>, find P.

Magnitude of a Vector, v |𝑣|= ∆𝑥 2 + ∆𝑦 2 If v = <a,b>, then |v|= 𝑎 2 + 𝑏 2 Example 2 Find the magnitude of the vector v represented by 𝑃𝑄 , 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 = < 𝑢 1 , 𝑢 2 > and v = < 𝑣 1 , 𝑣 2 >, the sum (or resultant) of the vectors is u + v = < 𝑢 1 + 𝑣 1 , 𝑢 2 + 𝑣 2 > The product of the scalar k and the vector u is ku = k < 𝑢 1 , 𝑢 2 > = <k 𝑢 1 ,𝑘 𝑢 2 >

Vector Operation Examples Example 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 = 𝒗 |𝑣| 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 Angles Using 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 Vector Find the components of a vector v with direction angle 135 degrees and magnitude 10. Ex. 6: Find Direction Angle of Vector Find 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 Dot 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 Results in a scalar Also known as the “inner product”

Dot Product The dot product or inner product of u = < 𝑢 1 , 𝑢 2 > and v = < 𝑣 1 , 𝑣 2 >is u · v = 𝑢 1 𝑣 1 + 𝑢 2 𝑣 2 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 u · u = |𝒖| 𝟐 0 · u = 0 u · (v + w) = u · v + u · w (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 = 𝒖∙𝒗 |𝒖||𝒗| and θ = 𝑐𝑜𝑠 −1 ( 𝒖∙𝒗 |𝒖||𝒗| )

Example: Finding the Angle Between Two Vectors Use an algebraic method to find the angle between the vectors u and v. u = <4, 1>, v = <-3, 2> 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 𝑝𝑟𝑜𝑗 𝑣 𝒖=( 𝒖·𝒗 |𝒗| 𝟐 )v