Section 1.1 Vectors The vector from point A ( x 1 , y 1 ) to the point B ( x 2 , y 2 ) is AB x 2 x 1 , y 2 y 1 run , rise . Note that infinitely many pairs of points result in the same vector. Scalar: In a discussion of vectors, ordinary numbers are called scalars. Components: The components of the vector a , b are the scalars a and b . A vector can be thought of as a set of directions for moving from point A to point B. Example: Find the vectors AB , BA and CD for the points A ( 1, 2 ), B ( 3 , 5 ), C ( 2 , 7 ) and D ( 6 ,10 ) . Length of a vector: The length of the vector a , b is a,b a b . 2 2 Note that AB is the distance from point A to point B. Scalar Multiplication: c a , b ca , cb . Note that BA AB . A unit vector is a vector of length 1. Example: Find a unit vector with the same direction as 3 , 4 . Then find two vectors of length 2 that are parallel to 3 , 4 . Adding Vectors: a 1 , b1 a 2 , b 2 a 1 a 2 , b1 b 2 . Examples: 1) Sketch v 1, 3 and w 4 , 1 . Sketch v w 2) Sketch a parallelogram with adjacent edges formed by the vectors v 1 , 3 and w 4 , 1 . What are the diagonals? 3) The parallelogram PQRS has adjacent edges PQ and PS for P (1, 2 ), Q ( 2 , 5 ), and S ( 5 ,1) . Find the vertex R. Basic Unit Vectors: The vectors i 1, 0 and j 0 ,1 are the basic unit vectors. Any vector a , b can be written as a i b j . Base Angle of a Vector: For a , b OA for O ( 0 , 0 ), A ( a , b ) Let be the angle between a , b OA and the positive x-axis. Then a,b a b cos i sin j . 2 2 Example: A 50 lb weight hangs from a ceiling by two strings. String 1 makes a 30 degree angle with the ceiling to the left and String makes a 45 degree angle to the right. The weight is at rest. Find the tensions in the strings.