Vectors Components and Magnitude

Objectives Write vectors in Component Format Find direction angles of vectors

A ball flies through the air at a certain speed and in a particular direction. The speed and direction are the velocity of the ball. The velocity is a vector quantity since it has both a magnitude and a direction. Vectors are used to represent velocity, force, tension, and many other quantities. A vector is a quantity with both a magnitude and a direction. Magnitude and Direction

A vector with initial point (0, 0) is in standard position and is represented uniquely by its terminal point (u 1, u 2 ). Standard Position If v is a vector with initial point P = (p 1, p 2 ) and terminal point Q = (q 1, q 2 ), then 1. The component form of v is v = q 1  p 1, q 2  p 2 2. The magnitude (or length) of v is ||v|| = x y (u 1, u 2 ) x y P (p 1, p 2 ) Q (q 1, q 2 )

Example: Find the component form and magnitude of the vector v with initial point P = (3,  2) and terminal point Q = (  1, 1). Example: Magnitude The magnitude of v is ||v|| = = = 5. =, 34  p 1, p 2 = 3,  2 q 1, q 2 =  1, 1 So, v 1 =  1  3 =  4 and v 2 = 1  (  2) = 3. Therefore, the component form of v is, v 2 v1v1

Example 2 Find the component form and magnitude of the vector v that has initial point (4,-7) and terminal point (-1,5). = = Magnitude of v

x y The direction angle  of a vector v is the angle formed from the x-axis counterclockwise to the ray along which v lies. x y v θ v θ Direction Angle

y x ɵ

4 Scenarios Must know the vector Quadrant so you calculate the correct direction angle. (See next slide)

Example 1

Homework WS 7-1