8-8 Chapter 8 Applications of Trigonometry 8.3 Vectors, Operations, and the Dot Product Basic Terminology Algeraic Interpretation of Vectors Operations with Vectors Dot Product and the Angle etween Vectors Basic Terminology Quantities that involve magnitudes, such as 45 l or 60 mph, can e represented y real numers called Other quantities, called, involve oth magnitude and direction for example, 60 mph east. A vector quantity can e represented with a directed line segment (a segment that uses an arrowhead to indicate direction) called a The length of the vector represents the of the vector quantity. When two letters name a vector, the first indicates the and the second indicates the of the vector. Two vectors are equal if and only if they have the same direction and the same magnitude. The sum of two vectors is also a vector. Vector addition is commutative. For every vector v there is a vector v that has the same magnitude as v ut opposite direction. Vector v is the of v. The sum of v and v has magnitude and is the Algeraic Interpretation of Vectors A vector with its initial point at the origin in a rectangular coordinate system is called a A position vector u with its endpoint at the point (a, ) is written a,, so u a,. Geometrically a vector is a directed line segment while algeraically it is an ordered pair. The numers a and are the and the, respectively, of vector u. The figure shows the vector u a,. The positive angle etween the x-axis and a position vector is the for the vector. In the figure, is the direction angle for vector u.
Section 8.8 Parametric Equations, Graphs, and Applications 8-9 Magnitude and Direction Angle of a Vector a, The magnitude (length) of vector u a, is given y the following. The direction angle satisfies tan, where a 0. a 2 2 u a EXAMPLE 1 Finding Magnitude and Direction Angle Find the magnitude and direction angle for u 3, 2. Horizontal and Vertical Components The horizontal and vertical components, respectively, of a vector u having magnitude u and direction angle are the following. That is, u a, u cos, u sin. a u cos and u sin EXAMPLE 2 Finding Horizontal and Vertical Components Vector w in the figure has magnitude 25.0 and direction angle 41.7. Find the horizontal and vertical components.
8-10 Chapter 8 Applications of Trigonometry EXAMPLE 3 Writing Vectors in the Form a, Write vectors u and v in the figure in the form a,. Vector Operations Let a,, c, d, and k represent real numers. Addition: a, c, d a c, d Scalar Multiplication: k a, ka, k Negative: If u a1, a2, then u a1, a2. Sutraction: a, c, d a, c, d a c, d EXAMPLE 5 Performing Vector Operations Let u 2, 1 and v 4, 3. See the figure. Find and illustrate each of the following. (a) u + v
() 2u Section 8.8 Parametric Equations, Graphs, and Applications 8-11 (c) 3u 2v i, j Form for Vectors If v a,, then v ai j, where i 1, 0 and j 0, 1. Dot Product The dot product of the two vectors u a, and v cd, is denoted u v, read u dot v, and given y the following: u v = a, c, d = ac + d The dot product of two vectors is a, not a vector. The dot product of two vectors can e positive, 0, or negative. EXAMPLE 6 Finding Dot Products Find each dot product. (a) 2, 3 4, 1 () 6, 4 2, 3 Properties of the Dot Product For all vectors u, v, and w and real numers k, the following hold. k k k (a) u v v u () u v w u v u w (c) u v w u w v w (d) u v u v u v (e) 0 u 0 (f) u u u 2
8-12 Chapter 8 Applications of Trigonometry Geometric Interpretation of Dot Product If is the angle etween the two nonzero vectors u and v, where 0 180, then the following holds. cos u v u v For angles etween 0 and 180,cos is positive, 0, or negative when is less than, equal to, or greater than 90, respectively. Reflect: What does the sign of the dot product of two vectors tell us aout the angle etween the two vectors? EXAMPLE 7 Finding the Angle etween Two Vectors Find the angle etween the two vectors. (a) u 3, 4 and v 2, 1 () u 2, 6 and v 6, 2