6.3-Vectors in the Plane

A vector is a quantity with both a magnitude and a direction. A ball flies through the air at a certain speed and in a particular direction. The speed and direction are the velocity of the ball. The velocity is a vector quantity since it has both a magnitude and a direction. Vectors are used to represent velocity, force, tension, and many other physical quantities.

Two vectors, u and v, are equal if the line segments A quantity with magnitude and direction is represented by a directed line segment PQ with initial point P and terminal point Q. P Q The vector v = PQ is the set of all directed line segments of length ||PQ|| which are parallel to PQ. Two vectors, u and v, are equal if the line segments representing them are parallel and have the same length or magnitude. v u Directed Line Segment

Scalar Multiplication Scalar multiplication is the product of a scalar, or real number, times a vector. Scalar Multiplication For example, the scalar 3 times v results in the vector 3v, three times as long and in the same direction as v. v 3v The product of - and v gives a vector half as long as and in the opposite direction to v. v - v

Geometric- Vector Addition u To add vectors u + v: 1. Place the initial point of u at the terminal point of v. 2. Draw the vector with the same initial point as v and the same terminal point as u. v u u v u + v

To subtract vectors u - v: Vector Subtraction Vector Subtraction v u To subtract vectors u - v: 1. Place the initial point of -v at the terminal point of u. 2. Draw the vector u  v from the terminal point of v to the terminal point of u. u u  v -v

1. The component form of v is v = q1  p1, q2  p2 A vector with initial point (0, 0) is in standard position and is represented uniquely by its terminal point (u1, u2). Standard Position x y (u1, u2) If v is a vector with initial point P = (p1 , p2) and terminal point Q = (q1 , q2), then 1. The component form of v is v = q1  p1, q2  p2 2. The magnitude (or length) of v is ||v|| = x y P (p1, p2) Q (q1, q2)

Example: Find the component form and magnitude of the vector v with initial point P = (3, 2) and terminal point Q = (1, 1). = , 3 4 - p1 , p2 = 3, 2 q1 , q2 = 1, 1 So, v1 = 1  3 =  4 and v2 = 1  ( 2) = 3. Therefore, the component form of v is , v2 v1 The magnitude of v is ||v|| = = = 5. Example: Magnitude

Example: Equal Vectors Two vectors u = u1, u2 and v = v1, v2 are equal if and only if u1 = v1 and u2 = v2 . Example: Equal Vectors Example: If u = PQ, v = RS, and w = TU with P = (1, 2), Q = (4, 3), R = (1, 1), S = (3, 2), T = (-1, -2), and U = (1, -1), determine which of u, v, and w are equal. Calculate the component form for each vector: u = 4  1, 3  2 = 3, 1 v = 3  1, 2  1 = 2, 1 w = 1  (-1), 1  (-2) = 2, 1 Therefore v = w but v = u and w = u. /

Operations on Vectors in the Coordinate Plane Let u = (x1, y1), v = (x2, y2), and let c be a scalar. 1. Scalar multiplication cu = (cx1, cy1) 2. Addition u + v = (x1+x2, y1+ y2) 3. Subtraction u  v = (x1  x2, y1  y2)

Examples: Operations on Vectors Examples: Given vectors u = (4, 2) and v = (2, 5) x y -2u = -2(4, 2) = (-8, -4) (4, 2) u 2u (-8, -4) u + v = (4, 2) + (2, 5) = (6, 7) u  v = (4, 2)  (2, 5) = (2, -3) x y x y (2, 5) (4, 2) v u (6, 7) (2, 5) (4, 2) v u u + v u  v (2, -3) Examples: Operations on Vectors

Unit Vectors Simply put, a unit vector is a vector which has a length of one unit. is a unit vector because its magnitude is: is a unit vector because its magnitude is:

A unit vector can be found by dividing a given vector by its magnitude: Ex. Find a unit vector in the direction of v = <-2, 5>

are special standard unit vectors in the positive X, Y direction. Ex. Let u be the vector with initial point (2, -5) and terminal point (-1,3). Write u as a linear combination of the standard unit vectors.

Ex. Let and Find 2u – 3v.

Direction Angle The direction angle  of a vector v is the angle formed by the positive half of the x-axis and the ray along which v lies. x y x y θ v v θ x y v (x, y) θ

Ex. If v = <3, 4>, what is the direction angle? y v (x, y) How could we determine the angle? θ If v = x, y , then tan  Ex. If v = <3, 4>, what is the direction angle? tan  = and  = 51.13