Parallel & Perpendicular Vectors in Two Dimensions
If we have cv, it is a scalar multiplied times a vector. What about a vector times a vector? Dot Product: it’s a number! (not a vector) Ex 1) Two Truths & a Lie Find the dot product. A) B) C) 29 7 0 should be 13 Perpendicular vectors have a dot product of 0 called orthogonal vectors.
We can utilize the Law of Cosines to find the angle between any two vectors. θ Ex 2) Find the measure of the angle between vectors
Parallel vectors have the same slope, they are scalar multiples of each other. watch out!
Example: Let a = (8, -4) and b = (2, 1) Show that a b = b a Find the angle between a and b to the nearest tenth of a degree Find a vector that is parallel to a. Find a vector that is perpendicular to a.
Ex 4) Determine the value of K for which each pair of vectors is parallel and the value of K for which they are perpendicular. Perpendicular: Parallel:
An important application of the dot product in physics is work done on a body through distance. Work = Force · displacement (vector) (vector) Ex 5) Determine the work done by a force of magnitude (newtons) in moving a box 20 m along a floor that makes an angle of 30° with . Give answers in newton-meters (N-m) (joules = newton-meters)
Properties of the Dot Product Norm Commutative Property Distributive Property Associative Property Scalar
Exit Slip We need to be able to tell if 2 vectors are parallel, perpendicular, or neither using the dot products. Choose two different options (between , , & N) Make up 2 questions of your own. Trade with a partner & solve theirs.