MATH 151 Engineering Mathematics I Spring 2018, WEEK 1 JoungDong Kim Week 1 Vectors, The Dot Product, Vector Functions and Parametric Curves. Section 1.1 Vectors Definition. A Vector is a quantity that has both magnitude and direction. A two-dimensional vector is an ordered pair a = a 1,a 2 of real numbers. The numbers a 1 and a 2 are called the components of a. a 1 is x component, a 2 is y component. Vector has an initial point and terminal point Note. If the initial point is origin, we call position vector. 6 5 4 3 2 1 6 5 4 3 2 1 1 1 2 3 4 5 6 2 3 4 5 6 y x 1
Given the points A(x 1,y 1 ) and B(x 2,y 2 ), the vector AB is x 2 x 1,y 2 y 1 Ex.1) Find the vector presented by the directed line segment with initial point A(1, 2) and terminal point B(4, 3). The length of the vector a = a 1,a 2 is a = a 2 1 +a 2 2 2
Vector Addition. If a = a 1,a 2 and b = b 1,b 2, then the vector a+b is defined by a+b = a 1 +b 1,a 2 +b 2 Multiplication of a vector by a scalar. If c is a scalar and a = a 1,a 2, then the vector ca is defined by ca = ca 1,ca 2 Ex.2) If a = 4,3 and b = 2,1, find a and the vectors a+b, a b, 3b, and 2a+5b. 3
If v has direction angle θ, the components of v can be computed using the formula v = v cosθ, v sinθ Ex.3) Find the component of the vector r given that; a) r = 2 and r makes an angle of 60 with the positive x-axis. b) r = 7 and r makes an angle of 150 with the positive x-axis. c) r = 1 2 and r makes an angle of 45 with the positive x-axis. 4
Basis Vector (Standard) i = 1,0, j = 0,1 Unit Vector is a vector with length one, a a. Ex.4) If a = 2,1, find a unit vector of a and a vector with length 3 in the direction of a. 5
Applications to Physics and Engineering A force is represented by a vector because it has both a magnitude and direction. If several forces are acting on an object, Resultant Force experienced by the object is the vector sum of the forces. Ex.5) John walks due west on the deck of a ship at 3mph. The ship is moving north at 22mph. Find the speed and direction of John relative to the surface of the water. 6
Ex.6) Two forces F 1 and F 2 with magnitudes 10 lb and 12 lb act on an object at a point P as shown in the figure. Find the resultant force F acting at P as well as its magnitude and its direction. 7
Ex.7) Suppose that a wind is blowing into the direction S45 E at a speed of 50 km/h. A pilot is steering a plane in the direction N60 E at an speed of 250 km/h. Find the true course (direction of the resultant velocity vector of the plane and wind) and ground speed(magnitude of resultant). 8
Section 1.2 The Dot Product The work done by a constant force F in moving an object through a distant d is W = Fd, but this applies only when the force is directed along the line on motion of the object. Suppose the constant force is a vector F = PR pointing in some other dirction. If the force moves the object from P to Q, then the displacement vector is D = PQ. The magnitude of the force applied in the direction of motion, that is PS = F cosθ Thus the work done W in moving the object is given by W = F D cosθ. 9
Definition. The Dot product of two nonzero vectors a and b is the number a b = a b cosθ where θ is the angle between a and b, 0 θ π. If either a or b is 0, we define a b = 0. Ex.8) If the vectors a and b have lengths 4 and 6, and the angle between them is π, find a b. 3 Ex.9) Find the work done by a force of 20 lbs acting in the direction N50 W in moving an object 4 feet due west. 10
Ex.10) a is perpendicular to b, a b is? Two vectors a and b are orthogonal if and only if a b = 0. Ex.11) a and b are parallel, a b is? 11
When we know the components of vectors, the dot product of a = a 1,a 2 and b = b 1,b 2 is a b = a 1 b 1 +a 2 b 2 Ex.12) Dot product of a = 2,4 and b = 3, 1. Ex.13) Find the values of x for which the vector x,5x and x, 10 are perpendicular. Ex.14) Find the values of x for which the vector 2,x and x 1,3 are parallel. 12
Ex.15) Find the angle between the vectors a = 2,2 and b = 5, 3. Ex.16) Find the angle between the vector 1,5 and 2,3. 13
Ex.17) A force with representation F = 3,8 moves an object along a straight line from the point (2,3) to the point (4,5). Find the work done if the distance is measured in meters and the magnitude of the force is measured in Newtons. Ex.18) A woman exerts a horizontal force of 65lb on a crate as she pushes it up a ramp that is 20ft long and inclined at an angle of 20 above the horizontal. Find the work done on the box. 14
Projections If S is the foot of the perpendicular from R to the line containing PQ, then the vector with representation PS is called the Vector Projection of b onto a and is denoted by proj a b. The Scalar Projcetion of b onto a. (also called the component of b along a) is defined to be the magnitude of the vector projection, which is the number b cos θ, where θ is the angle between a and b, denoted by comp a b. The Scalar Projection of b onto a: The Vector Projection of b onto a: proj a b = comp a b = a b a ( a b a ) a a = a b a 2 a Ex.19) Find the vector and scalar projection of 4,8 onto 2,1. 15
Definition: Given the nonzero vector a = a 1,a 2, the orthogonal complement of a is the vector a = a 2,a 1. Ex.20) Find the orthogonal complement of a = 1,4. Graph both a and a on the same axis. 6 5 4 3 2 1 6 5 4 3 2 1 1 1 2 3 4 5 6 2 3 4 5 6 y x Ex.21) Find two unit vectors perpendicular to 2, 3. 16
Ex.22) Find the distance from the point P(2,1) to the line y = 2x+1. 17
Section 1.3 Vector Functions. Parametric Curves We call x = f(t) and y = g(t) parametric equations where t is the parameter. As t varies over its domain, we get a collection of points (x,y) = (f(t),g(t)) which traces out the parametric curves. Ex.23) Let x = t 3, y = 2t 1, Find x and y at t = 0, t = 1, t = 2. Degree of t is one, means this parametric equation represents straight line. And also we could combine these parametric functions as Cartesian equation. Ex.24) Let x = 1 2t, y = 2+3t, 3 t < 3, find the Cartesian equation. 18
Ex.25) x = t+1, y = t 2 4. Ex.26) x = 2sinθ, y = 3cosθ. Ex.27) x = sint, y = csct, π 6 t < π 2. 19
Vector Function We call r(t) = x(t),y(t) a Vector Function. Ex.28) Sketch the following curves described by the vector function. Include the direction of the curve as t increases. a) r(t) = t 1,2 3t. b) r(t) = 2+cost,1+sint. 20
Vector Equation of Line A vector equation of the Line passing through the point r 0 = (x 0,y 0 ) and parallel to the vector v = v 1,v 2 is given by r(t) = r 0 +tv From this vector equation, we can obtain the parametric equations of the line as follows; r(t) = r 0 +tv = x 0,y 0 +t v 1,v 2 = x 0 +tv 1,y 0 +tv 2 Parametric equations of the line that passes through the point (x 0,y 0 ) and is parallel to the vector v 1,v 2 are given by x(t) = x 0 +tv 1, y(t) = y 0 +tv 2 Ex.29) Find the vector equation of the line parallel to the vector 1,4 and passing through the point ( 1,5). Note. If v is parallel to the line, then any multiple of v is also parallel to the line and can be used to obtain a vector equation or parametrized equations of a line. 21
Ex.30) Find parametric equations for the line with slope 4 3 and passing through the point (2, 5). Ex.31) Find the vector equation of the line passing through the points (1,2) and ( 1,4). 22
Ex.32) Consider the line 2x+3y = 5. a) Find a vector parallel to the line. b) Find a vector perpendicular to the line. Ex.33) Anobjectismovinginthexy-planeanditspositionaftertsecondsisgivenbyr(t) = t+4,t 2 +2, a) Find the position of the object at t = 2. b) At what time does the object reach the point (7,11)? c) Eliminate the parameter to obtain a cartesian equation. 23
Ex.34) Consider the lines r(t) = 4+2t,5+t and s(w) = 2+3w,4 6w. Determine whether the lines are parallel, perpendicular or neither. If they are not parallel, find the intersection point. 24