Introduction to Vectors

1. Introduction to Vectors Section 12-5

2. Many of the quantities you work with in mathematics, such as those representing area, volume, and money in a bank account, are measures, or counts. • Other quantities involve both a measure and a direction. For example, distances often have directions associated with them. Other examples include velocity and acceleration. • Directed quantities like these are often represented by vectors. • A vector can be thought of as a directed line segment. It has a length, or magnitude, and a direction.

3. Numbers without an associated direction are called scalars.

4. Vectors can be represented in several ways. One way gives the magnitude and the angle the vector makes with the positive x-axis. This is called the polar form of the vector. For example, the polar form represents a vector 2 units long directed at an angle of 60° counterclockwise from the positive x-axis.

5. Another way to describe a vector is to give the horizontal and vertical change from the tail to the head. This is called the rectangular form of the vector. • When the tail of a vector is at the origin, then its rectangular form is the same as the coordinates of its head. The rectangular form of the vector at right would be designated as

6. Vectors are equivalent if they have the same length and point in the same direction. The location of the vector on the coordinate plane doesn’t matter. • All of the vectors in the picture at right are equivalent. • Each could be described by either or .

7. As you can add and subtract numbers without direction, or scalars, you can add and subtract vectors. • Vector addition can be accomplished geometrically by placing two vectors in a “tip to tail” arrangement. The sum, or resultant vector, is the vector that connects the tail of the first vector to the head of the final vector.

8. When writing by hand, you use arrows above the letters to refer to vectors. In printing, as in this book, vectors are often designated by boldface type without the arrows, as in a + b = c.

9. Vector Addition and Subtraction • On graph paper, draw a set of axes and the vector a= < 2, 3>. Remember to draw an arrowhead at the head, or tip, of the vector. • Add the vector b= < 4, 1> to a. Draw b so that the tail of b starts at the tip of a. The tip of b should be 4 units to the right and 1 unit up from its tail. Don’t forget the arrowhead at the tip.

10. Draw the sum, or resultant vector, c. What is its rectangular form? • Repeat Steps 1–3 to complete these vector sums: • b + a • d + e • b + f • a + e

11. Look at the rectangular form of the resultant vectors. Complete the following definition of vector addition for vectors in rectangular form.

12. Subtracting a number is the same as adding its opposite. It’s the same for vectors. • The opposite of vector b is called -b. It has the same magnitude as b, but it points in the opposite direction. The difference a - b is the same as the sum a + -b. • Draw a representation of each difference. • a - b • b - a • d - e • e - f

13. Based on your drawings, complete the following definition of vector subtraction for vectors in rectangular form.

14. Create a conjecture about multiplying a vector by a scalar (number). • For example, what would it mean to multiply 2 · a? (Hint: This is the same as adding a + a.) • Complete the following definition of scalar multiplication.

15. The magnitude (length) of a vector is symbolized by placing the vector name inside vertical bars, like an absolute-value sign. Find the magnitudes of a and b, then complete the following definition of the magnitude of a vector in rectangular form.

16. Example A • Ernie drives a taxi in a large city in which the streets are laid out in a square grid. One night he starts out from the garage and travels along this route: • 5 blocks east and 2 blocks north • 3 blocks west and 7 blocks north • 6 blocks east and 8 blocks south • 9 blocks east and 12 blocks north • 10 blocks west and 4 blocks south • Where is Ernie relative to the garage at the end of this trip?

17. Example A • Represent the first trip segment as a vector: Five blocks east and 2 blocks north is the same as the vector <5, 2>. • Similarly, the other trips can be represented by the vectors <3, 7>, <6, 8>, <9, 12>, and <10, 4>. • Finding the sum of all five vectors gives the final position: <5-3+6+9-10, 2+7-8+12-4> <7,9>

18. When traveling in a city with streets and buildings, it may be convenient to consider the rectangular form of the vector. But to represent movement in open space, you may want to use polar form to show the distance and the direction (angle) of the movement. You can use trigonometry to convert from one form to the other.

19. Example B • Convert the rectangular form, <4, 9>, of a vector to its polar form. • We need to find both the length, or magnitude, of the vector and its angle with the horizontal. Draw a diagram. • The vector is the hypotenuse of a right triangle with legs of lengths 4 and 9. • To find its magnitude, use the Pythagorean Theorem.

20. Convert the rectangular form, <4, 9>, of a vector to its polar form.

21. Convert the rectangular form, <4, 9>, of a vector to its polar form. To find the vector’s angle, you can use the inverse tangent: So the polar form of the vector is

22. Example C • A ship leaves port and travels 47 miles on a bearing of 28° to get out of a bay. It then turns and travels 94 miles on a bearing of 137° to reach its destination port. • Find the ship’s distance and bearing from port.

23. Example C • You can add the two vectors to obtain a single vector from the initial port to the destination, but you will first need to use trigonometry to convert the vectors from polar to rectangular form.

24. Example C • Change the angles to measure from the x-axis.

25. Example C

26. Example C

27. Example C The vector has an angle, so its bearing is 90+17.55o or 107.55°. The ship is about 90 miles from port at a bearing of about 108°.