SUPPLEMENT I. Example. Graph the vector 4, 3. Definition. Given two points A(x 1, y 1 ) and B(x 2, y 2 ), the vector represented by # AB is # AB =,

Size: px

Start display at page:

Download "SUPPLEMENT I. Example. Graph the vector 4, 3. Definition. Given two points A(x 1, y 1 ) and B(x 2, y 2 ), the vector represented by # AB is # AB =,"

Avis Wilson
5 years ago
Views:

1 SUPPLEMENT I 1. Vectors Definition. A vector is a quantity that has both a magnitude and a direction. A twodimensional 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. Example. Graph the vector 4, Definition. Given two points A(x 1, y 1 ) and B(x 2, y 2 ), the vector represented by # AB is # AB =, The magnitude or length of a vector a = a 1, a 2 is a = Example. For the points A(2, 3) and B(5, 6), find # AB and # BA. Example. Find the length of each of the following vectors. A) a = 4, 7 B) b = 0, 0

2 M151 Notes by R.G. Lynch, Texas A&M Supplement I, Page 2 of 10 Definition. If c is a scalar, and a = a 1, a 2 and b = b 1, b 2, then we define (Vector Addition) a + b =, (Scalar Multiplication) ca =, The magnitude of ca =. The difference a b =. Example. Compute the following for a = 4, 6 and b = 7, 1 A) a + b = B) 3a + 2b = C) a 3b = Definition. Two vectors a and b are said to be parallel if a = cb for some scalar c. Definition. A unit vector is a vector with length 1. The unit vectors i = 1, 0 and j = 0, 1 are referred to as the standard basis vectors. We can then write a = a 1, a 2 =. Example. Find a vector in the same direction as 3, 4 having length 7.

3 M151 Notes by R.G. Lynch, Texas A&M Supplement I, Page 3 of 10 Example. A pilot is flying in the direction of N60 W at an airspeed of 180 km/h. A) Find the velocity vector. B) Find true course and ground speed of the plane if there is a wind blowing in the direction of N45 E at 36 km/h.

4 M151 Notes by R.G. Lynch, Texas A&M Supplement I, Page 4 of 10 Example. A 75-lb weight hangs from two wires as shown. Find the tensions (forces) T 1 and T 2 in both wires and their magnitudes

5 M151 Notes by R.G. Lynch, Texas A&M Supplement I, Page 5 of The Dot Product So far we have added vectors and multiplied them by scalars. Can we multiply them in some meaningful way? Definition. The dot product of two nonzero vectors a and b is the number a b = a b cos θ where θ is the angle between a and b satisfying 0 θ π. If either a or b is zero, then we define a b = 0. Example. Find a b if a = 7, b = 9, and the angle between them is θ = π 6. Definition. The work done by a force F in moving an object from a point P to a point Q, or with displacement D = P # Q is given by W = F D. Example. A crate is hauled 5 m up a ramp an angle of 30 to the ground, pushed with a constant horizontal force of 14 N. Find the work done. Definition. We can alternatively define the dot product of two vectors a = a 1, a 2 and b = b 1, b 2 as a b = a 1 b 1 + a 2 b 2. Optional exercise. Check that these are equal. Dot Product properties. If a, b, and c are vectors and m is a scalar, then a a = a 2 a b = b a 0 a = 0 a (b + c) = a b + a c (ca) b = c(a b) = a (cb) Definition. Two nonzero vectors a and b are said to be orthogonal if a b = 0. The orthogonal compliment of a = a 1, a 2 is a = a 2, a 1.

6 M151 Notes by R.G. Lynch, Texas A&M Supplement I, Page 6 of 10 Example. Are the vectors a = 3, 4 and 5, 2 orthogonal? Question. If two nonzero vectors are orthogonal, what s the angle between them? Example. What value(s) of x will make x, 4 and x, 7x orthogonal? Example. Find the angle between a = 3i + 5j and b = 4i + 2j.

7 M151 Notes by R.G. Lynch, Texas A&M Supplement I, Page 7 of 10 Definition. The vector projection of b = P # R onto a = P # Q, denoted as proj a b is the vector P # S. The scalar projection is the magnitude of the vector projection, also called the the component of b along a, and is denoted by comp a b. Example. Find the scalar projection and the vector projection of b = 3, 2 onto a = 4, 6. Example. Find the vector projection of b = 3, 2 onto i.

8 M151 Notes by R.G. Lynch, Texas A&M Supplement I, Page 8 of Parametric Equations and Vector Functions For some applications, such as modeling the path of an object, it is more convenient to use vector functions rather than Cartesian functions of the form y = f(x). Definition. A curve x = x(t), y = y(t) is called parametric curve with where the variable t is called the parameter. For each t, we can view the point (x(t), y(t)) on a parametric curve as the end point of the vector r(t) = x(t), y(t) = x(t) i + y(t) j called the position vector. The position vector is an example of a vector function. Example. Graph the vector function r(θ) = 2 sin θ, 2 cos θ, with π/2 θ π/2. Example. Suppose that r(t) = t 2 i + (t 3)j with 3 t 3. A) Is the point (4, 1) on the graph of r(t)? B) Sketch the graph of r(t)

9 M151 Notes by R.G. Lynch, Texas A&M Supplement I, Page 9 of 10 C) Eliminate the parameter to find the Cartesian equation of the curve. Example. Find the Cartesian equation of x = sin(2θ), y = sin(θ). Definition. Let L be a line on which a point P 0 (x 0, y 0 ) lies. The vector equation of L is r(t) = r 0 + tv where r 0 = x 0, y 0 is the vector formed by the origin and P 0, and v = a, b is a vector parallel to L. The parametric equations of a line are x(t) = y(t) = Example. Find the vector equation of the line passing through the points ( 2, 5) and (4, 3).

10 M151 Notes by R.G. Lynch, Texas A&M Supplement I, Page 10 of 10 Example. Find the vector equation of y = 4x 7. Example. Determine whether the lines L 1 and L 2 are parallel, perpendicular, or neither. If they are not parallel, find their point of intersection. L 1 : L 2 : r 1 (t) = ( 4 + 4t)i + (1 + t)j r 2 (s) = (2 + 2s)i + (2 8s)j

Vector Supplement Part 1: Vectors

Vector Supplement Part 1: Vectors A vector is a quantity that has both magnitude and direction. Vectors are drawn as directed line segments and are denoted by boldface letters a or by a. The magnitude