C H. 6 – A DDITIONAL T OPICS IN T RIGONOMETRY 6.3 – Vectors!

V ECTOR !

V ECTORS ! Vector = A quantity with both magnitude and direction It has an initial point (P) and an end point (Q) Component form : the traveled by the vector Think of it as the end coordinates of the vector if it started at the origin! For PQ, just do end point minus initial point, or Q – P: So the component form is. P (0, 1) Q (3, 2) - P (0, 1) Q (3, 2)

V ECTORS ! Magnitude : length of the vector The magnitude of v is denoted || v || To find it, use distance formula! Ex: What is the magnitude of a vector starting at (2, -2) and ending at (3, 5)? Answer: Ex: Find ||u|| if u =. Answer:

L ET U = AND V = Vector Addition: u + v = Scalar Multiplication: k u = Ex: Let w = and z =. Find w – 2 z. Find one coordinate at a time! For the 1 st coordinate, (-3) – 2(5) = -13 For the 2 nd coordinate, (4) – 2(2) = 0 Answer: u v v

Draw vector sums and differences end to start! Ex: Given u and v in the figure below, draw the following: u + v v – u u + 2v u v u + v u v - u

Draw vector sums and differences end to start! Ex: Given u and v in the figure below, draw the following: u + v v – u u + 2v u v v

W HAT IS THE MAGNITUDE OF U ? R OUND TO THE NEAREST TENTH

U = AND V =. 0.5 U + 2 V = ?

U NIT VECTORS Unit Vector = a vector of magnitude 1 = A vector divided by its magnitude is a unit vector! Ex: Find a unit vector in the same direction as u. Divide u by its magnitude! You may also see a vector written in i, j form, where i is the x-coordinate and j is the y-coordinate of the vector:

V ECTOR A NGLES Ex: Find the direction angle of v = 3 i + 7 j. Angles are always measured counterclockwise from the positive x-axis! Draw the vector and make a right triangle! Use trig to find the angle! Ex: Find the direction angle of u = 17 i – 6 j. But we want the counterclockwise angle, so… v 3 7 u 17 6 θ θ

V ECTOR A NGLES Ex: A football is thrown with a speed of 52 mph at a 35° angle of elevation. Express this velocity in vector form. Draw the vector and make a right triangle, then use trig! So the answer is. 52 i j 35°