Vectors TS: Explicitly assessing information and drawing conclusions. Warm Up: a)What are the coordinates of A and B b)What is the distance between A and B?

Equivalent Directed Line Segments B A C

Component form of a Vector The component form of a vector with initial point P (p 1, p 2 ) and terminal point Q (q 1, q 2 ) is given by The magnitude of v is given by SPECIAL VECTORS: If ||v|| = 1, v is a unit vector. If ||v|| = 0, v is the zero vector. v1v1 v2v2 ||v|| P Q

Ex : Find the component form and magnitude of the vector shown, then drawn an equal vector whose terminal point is (5, 2)

Vector Operations: Scalar Multiplication Given v find: 1)2v 2)-v 3)0.5v

Vector Operations: Addition Given u and v Find u + v u v

Vector Operations: Subtraction Find u - v u v

Example: Multiple Operations Find u – 2v u v

Properties of Vector Addition and Scalar Multiplication Let u, v, and w be vectors and let c and d be scalars. Then the following properties are true. 1)u + v = v + u 2)u + 0 = u 3) c(du) = (cd)u 4) c(u + v) = cu + cv 5)||cv|| = |c| ||v|| 6)(u + v) + w = u + (v + w) 7)u + (-u) = 0 8)(c + d)u = cu + du 9)1(u) = u 10) 0(u) = 0

Unit Vectors To find a unit vector divide the vector, v, by its magnitude. This will have the same direction as the vector, v, but it’s magnitude (or length) will be 1.

Example: Find a unit vector going in the same direction as. Verify it is a unit vector.

Writing a vector as a Linear Combination of Unit Vectors, & Unit Vectors: = and = Linear Combination of Unit Vectors v = = v 1 + v 2

Example: If u is a vector with initial side (2, -5) and terminal side (-1, 3), write u as a linear combination of the standard unit vectors and