Supplement: 1.1 Introduction to Vectors and Vector Functions

Transcription

1 Math 151 c Lynch 1 of 6 Supplement: 1.1 Introduction to Vectors and Vector Functions The term vector is used by scientists to indicate a quantity (such as velocity or force) that has both magnitude and direction. Definition. 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. Note. A representation of the vector a = a 1, a 2 is a directed line segment AB from any point A(x, y) to the point B(x + a 1, y + a 2 ). A particular representation of a is the directed line segment OP from the origin to the point P (a 1, a 2 ), and a 1, a 2 is called the position vector of the point P (a 1, a 2 ). Proposition. Given the points A(x 1, y 1 ) and B(x 2, y 2 ), the vector a with representation AB is a = x 2 x 1, y 2 y 1. Example 1. Find the vector represented by the directed line segment with initial point A(5, 7) and terminal point B(3, 1). Theorem. The magnitude or length of the vector a = a 1, a 2 is a = a a2 2 The length of the vector from points A(x 1, y 1 ) to B(x 2, y 2 ) is AB = (x2 x 1 ) 2 + (y 2 y 1 ) 2 Remark. The only vector with length zero is the zero vector 0 = 0, 0.

2 Math 151 c Lynch Supplement 1.1 Vectors 2 of 6 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 Remark. Looking at the graphs below, you can see why the definition of vector addition is sometimes referred to as the Triangle Law or the parallelogram law. 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 Note. If c > 0, then the vectors a and ca point in the same direction. If c < 0, then the vectors a and ca point in opposite directions. The length of the vector ca is c times the length of a, in other words ca = c a. Two vectors a and b are called parallel if b = ca for some scalar c. Vector Subtraction. By the difference a b of two vectors, we mean a b = a+( b) so if a = a 1, a 2 and b = b 1, b 2, then a b = a 1 b 1, a 2 b 2 Example 2. If a = 2, 4 and b = 3, 2, find: (a) a = (b) a + b = (c) a b = (d) 4b = (e) 2a + 5b =

3 Math 151 c Lynch Supplement 1.1 Vectors 3 of 6 Properties of Vectors. If a, b, and c are vectors and c and d are scalars, then 1. a + b = b + a 2. a + (b + c) = (a + b) + c 3. a + 0 = a 4. a + ( a) = 0 5. c (a + b) = ca + cb 6. (c + d)a = ca + da 7. (cd)a = c(da) 8. 1a = a. 9. ca = c a Definition. The two vectors i and j are defined as i = 1, 0 and j = 0, 1 The vectors i and j have length 1 and point in the direction of the positive x- and y-axes, respectively. We refer to these as standard basis vectors. Theorem. For any vector a = a 1, a 2, we can write the vector a as a = a 1 i + a 2 j. So any vector can be written as scalar multiples of the vectors i and j added together. Example 3. If a = 2i 3j and b = 5i + 6j, express the vector 2a 4b in terms of i and j. Definition. A unit vector is a vector whose length is 1. Theorem. If a 0, then the unit vector that has the same direction as a is u = 1 a a = a a Example 4. Find the unit vector in the direction of the vector 4i + 7j.

4 Math 151 c Lynch Supplement 1.1 Vectors 4 of 6 Example 5. Find the vector that has the same direction as 3, 6 but has length 7. Example 6. If v lies in the fourth quadrant and makes an angle π/3 with the negative y-axis and v = 7, find v in component form. Example 7. Suppose a canon is aimed at an angle 25 above the ground and a cannonball is fired at a velocity of 300 m/s. Find the horizontal and vertical components of the velocity vector.

5 Math 151 c Lynch Supplement 1.1 Vectors 5 of 6 Remark. A force is represented by a vector because it has both a magnitude (measured in pounds or newtons, N) and a direction. If several forces are acting on an object, the resultant force experiences by the object is the vector sum of these forces. Example 8. Suppose that two forces F 1 and F 2 with magnitudes of 5 lb and 7 lb, respectively, act on an object (a diagram will be given in lecture). (a) Find the resultant force F acting on the object. (b) Find the magnitude of F and the angle it makes with the positive x-axis. Example 9. Suppose that the current in the ocean is flowing in a direction S30 W at a speed of 4 km/h. A ship is sailing in a direction N60 W at a speed in still water of 20 km/h. The true course, or track, of the ship is the direction of the resultant of the velocity vectors of the ship and the ocean. The speed of the ship is the magnitude of the resultant. Find the true course and the speed of the ship. (Round your answer to once decimal place.)

6 Math 151 c Lynch Supplement 1.1 Vectors 6 of 6 Example 10. A 100-kg weight hangs from two wires (a diagram will be given in lecture). Find the tensions (forces) T 1 and T 2 in both wires and their magnitudes. (Use 9.8 m/s 2 for the acceleration due to gravity. Round your answers to two decimal places.)