5.4 Vectors terminal point initial point Q v S w P R A geometric vector in 2-dimensional space is a directed line segment with initial point P and terminal point Q. A vector such as this would be denoted as v = PQ The magnitude of this vector is the length of the line segment from P to Q. This is represented by ||v||. A vector with P = Q is called a zero vector because the magnitude = 0. Vectors with the same magnitude and same direction are equal. v =PQ = w = RS Notice that even though v and w have different initial points and different terminal points, they can still be equal because they have the same magnitudes and directions. terminal point Q P initial point Q v S w P θ R θ

Vector addition is commutative, which means v + w = w+v Adding Vectors You can add vectors v and w be repositioning them so that the terminal point of v coincides with the initial point of w. This results in a new vector, v+w. w v v v+w w Notice that the new vector v+w has the same initial point as v and the same terminal point as w (after they’ve been repositioned) Vector addition is commutative, which means v + w = w+v Vector addition is also associative, which means u+(v+w) = (u+v)+w w w+v v+w v v u+v+w = u+(v+w) = (u+v)+w w u+v u

Vector subtraction is done by adding opposite vectors. That is, u – v = u + (-v), where –v is the vector having the same magnitude as v, but whose direction is opposite to v. -w -s v Notice that r – s = r +(-s) v-w r-s r w – w = 0 w -w Multiplying Vectors by Numbers. When dealing with vectors, we refer to real numbers as scalars. If a is a scalar and v is a vector, the scalar product of av is defined as follows: 1. If a>0, the product av is the vector whose magnitude is a times the magnitude of v and whose direction is the same as v. If a<0, the product av is the vector whose magnitude is |a| times the magnitude of v and whose direction is opposite that of v. If a=0 or if v=0, then αv = 0. Properties: 0v =0 1v = v -1v = -v (a+b)v = av + bv a(v+w) =av + aw a(bv) =(ab)v -v w v 2w 2v

Example 1 v u w b) c) 2v 3w 2v+3w u 2v-w+u -w 2v-w NOWYOU TRY #13 2v 2 units up u 3 units up w 3 units right 2 units down 2 units right 1 unit right 2v 3w 2v+3w u 2v-w+u -w 2v-w NOWYOU TRY #13 2v

A unit vector u is a vector u for which ||u|| = 1 i and j are special unit vectors that lie on the x-axis and y-axis, respectively. i = <1,0> and j = <0,1> Any nonzero vector can be made into a unit vector by dividing itself by its magnitude. This is a unit vector that as the same direction as v, but a magnitude of 1. Position Vectors An algebraic vector v is represented as v = , where a and b are scalars (real numbers) called the components of v. An algebraic vector may also be written in the form v = ai + bj. The a denotes the x-axis (horizontal) component and the b denotes the y-axis (vertical) component of v. (Notice this is an entirely different notation than the complex plane.) A position vector is an algebraic vector whose initial point is the origin. Theorem: Suppose v is a vector with initial point P1 = (x1,y1), not necessarily the origin, and terminal point P2 = (x2,y2). If v = P1P2, then v is equal to the position vector P2(5,6) P1(1,3) (4,3)=(5-1,6-3)

ai + bj vector form addition subtraction, and scalar multiplication. Let v = a1i + b1j and w= a2i + b2j You do #33 and #39 If the direction of a vector, v, is given by θ, which is the angle between x-axis (or unit vector, i) and v, then v can be expressed in ai + bj form in term of its magnitude and direction as v = ||v||[(cosθ)i + (sinθ)j] = ||v||(cosθ)i + ||v||(sinθ)j v θ 1 sin θ cos θ (cosθ, sinθ) ||v||

Applications A velocity vector whose magnitude is an object’s speed and direction is the direction object is moving in. A force vector is a vector whose magnitude is the amount of force acting on an object and the direction is the direction of the force (i.e., gravity is a force acting in the downward direction (-j)). Example 6 A ball is thrown with an initial speed of 25 miles per hour in a direction that makes an angle of 30° with the positive x-axis. Express the velocity vector, v, in terms of I and j. What is the initial speed in the horizontal direction (i)? What is the initial speed in the vertical direction (j)? The speed is the magnitude, so ||v|| = 25. θ = 30° . The initial speed in the horizontal direction in the horizontal component of v. Let’s put v in ai+bj form in terms of θ . v = 25[(cos 30°)i + (sin 30°)j]

F3 = Force of gravity down = <0, -1200> Example 7 A box of supplies that weights 1200 pounds is suspended by two cables attached to the ceiling. What is the tension in the two cables? 30° 45° F1 150° F2 30° 45° F1 and F2 are the forces of tension in the cables holding the box in equilibrium. 1200 pounds F3 = Force of gravity down = <0, -1200> For static equilibrium, the sum of the force vectors must equal zero. F1 + F2 + F3 = 0 = <0,0> Therefore the sum of the i components will be zero, and the sum of the j components will be zero. We must find the magnitude and direction of each of these vectors. We know the magnitude and direction of F3. || F3 || = 1200, and the direction is straight down (270°). Now we must find the magnitude and directions of F1 and F2. F1 = F2 =

HOMEWORK p.382 #7-13 odd, #25-49 odd, 61,65