Mr. Rommel South Salem HS Vectors Parallel and Perpendicular Vectors Dot Product

Mr. Rommel South Salem HS Parallel Lines In Elementary school you probably learned that two lines that never cross are called parallel lines. Though the actual definition are two lines that equidistant apart, the first definition usually is enough. Find an example of two lines that never cross that are not parallel.

Mr. Rommel South Salem HS Parallel Vectors Two vectors are equal if they have the same direction and magnitude. Two vectors are parallel if they have the same direction. This can be represented by v 1 = kv 2 Having the same direction on a coordinate plane can be represented by having the same slope.

Mr. Rommel South Salem HS Perpendicular Vectors Two lines are perpendicular if they intersect at a 90° angle Two vectors are perpendicular if the lines that contain them are perpendicular. The slopes of perpendicular lines are opposite reciprocals (m 1 )(m 2 )= -1

Mr. Rommel South Salem HS Perpendicular Vectors If we use the vectors (x 1, y 1 ) and (x 2, y 2 ) to represent two perpendicular vectors we can write: So if the sum of the products of the x and y coordinates is 0 then the vectors are perpendicular

Mr. Rommel South Salem HS We will use this quantity throughout our work with vectors so in order to make it easier to write we give it a special name, dot product. The properties of the dot product are similar to multiplication with a few differences: u v = v u u u = |u| 2 k(u v) = ku v u (v + w) = u v + u w

Mr. Rommel South Salem HS Angle between two vectors Using the dot product we can find the angle formed by two vectors. (This proof is on page 442 in your book) To do this we take vector u and v and find the difference between them v u u - v

Mr. Rommel South Salem HS Using the distance formula we can show that: v u u - v

Mr. Rommel South Salem HS v u u - v From the law of cosines: Where 0° ≤ θ ≤ 180°

Mr. Rommel South Salem HS Example To the nearest tenth of a degree find the measure of the angle between the vectors u=(1,3) v=(-8,5) 76.4°

Mr. Rommel South Salem HS Questions? Assignment: Pg all