VECTORS IN A PLANE Pre-Calculus Section 6.3

CA content standards: Trigonometry 12.0 Students use trigonometry to determine unknown sides or angles in right triangles. 13.0 Students know the law of sines and the law of cosines and apply those laws to solve problems. 14.0 Students determine the area of a triangle, given one angle and the two adjacent sides. 19.0 Students are adept at using trigonometry in a variety of applications and word problems.

OBJECTIVES Represent vectors as directed line segments Write vectors in component form Add and subtract vectors and represent them graphically Perform basic operations on vectors using scalars Write vectors as linear combinations of i and j Find the direction angle of a vector Apply vectors to real-world problems

Vector Directed line segments Named by initial point and terminal point (like a ray, in geometry) Ex: PQ P Q

Vectors have direction and magnitude Magnitude = length Given the endpoints of a vector use the distance formula to find its magnitude

Vectors with the same direction and magnitude are equal. Vectors can also be named using a single, bold, lowercase letter Ex: u=PQ

Given P=(0,0) Q=(3,4) R=(4,3) S=(1,2) T=(-2,-2) a=PQ, b=RP, c=ST, d=QP which vectors are equivalent?

component form Standard position – initial point at origin Component form – use terminal point to refer to vector v vy vx

Zero vector, 0 = <0,0> Unit vector

Component form: general position Remember equal vectors are determined by direction and magnitude – not location Rewrite in standard position

vector operations: scalar multiplication Scalar – number To multiply a vector by a scalar – multiply each component by that scalar ex

Vector operations: addition To add vectors, add their components Ex:

vector operations: addition Visually, vectors can be added using the parallelogram law Join vectors tail to head Resultant vector is diagonal of parallelogram

ex Visually and algebraically find

Unit vectors Remember a unit vector is any vector with magnitude of 1 To find a unit vector in the direction of a vector v, divide the vector by its magnitude Ex. Find the unit vector in the direction of <5,-2>

Unit vectors A vector can be written in terms of a directional unit vector and its magnitude Write in terms of the unit vector w/ the same direction

standard unit vectors Horizontal unit vector Vertical unit vector j i

ALL vectors in a plane can be written as a combination of i and j Ex. W has an intial point at (6,6) and terminal point (-8,3) write it as a combination of i and j W in component form is <-14, -3> As a combination of i and j, W = -14i – 3j

direction angles and vectors Direction angle is from the positive x axis. Use right triangle trig. v vy θ vx

Write each vector in component form 7 8 300° 30°

Write the magnitude and direction angle for each vector <-5,5>