6-3: VECTORS IN THE PLANE Various quantities in geometry and physics such as area, time and temperature can be measured in a single real number. However, quantities such as force and velocity have both a magnitude and direction that need to be characterized by more than a real number Vectors * directed line segments with the same magnitude and direction are equivalent -they have the same distance and same slopes! * represented by bold-faced, lowercase letters Example: u is the line segment from P=(0,0) to Q=(3,2) and v is the line segment from R=(1,2) to S=(4,4). Show that u=v. Determine magnitude: (distance formula) Determine direction: (slope) Component Form of a Vector v1 ,v2 Find the component form and magnitude of a vector v that has initial point (4, -7) and terminal point (-1, 5). Solution: v1 change in the x values If v= 5,12 v2 change , the magnitude of v is v (5) 2 (12) 2 169 13 in y values Vector Operations Let v= 2,5 and w= 3,4 , and find each of the following vectors. a. 2v b. w – v c. v + 2w d. 2v - 3w Unit Vectors *determine unit vectors by dividing v by its magnitude Find a unit vector in the direction of v= 2,5 . v v 2,5 (2) 2 (5) 2 v . Linear Combinations of Unit Vectors Another way to represent vectors in equation form 2,5 = -2i + 5j Operations with linear combinations work just like combining like terms Example: u= -3i + 8j and v= 2i – j find 2u – 3v.