Distance in the Plane

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Adam Caldwell
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1 Distance in the Plane The absolute value function is defined as { x if x 0; and x = x if x < 0. If the number a is positive or zero, then a = a. If a is negative, then a is the number you d get by erasing the minus sign in front of it. Thus, 5 = 5, 17 = 17, 0 = 0, 2 = 2, and 9 = 9. Because it s the job of the absolute value function to erase the minus sign in front of those numbers that have them, the absolute value of a number x is the same as the absolute value of its negative, x. Written with math symbols x = x As examples, 4 = 4 = 4 and 10 = 10 = ( 10). Whether you square a number or its negative, you ll get the same result. That is, x 2 = ( x) 2. And since x equals either x or x, depending on whether x is negative, we know that x 2 = x 2 As examples, 3 2 = 3 2 and 7 2 = 7 2 = ( 7)

2 Distance in R The distance between the two numbers a, b R is a b. Examples. The distance between the numbers 5 and 3 is 5 3 = 2 = 2. The distance between the numbers 3 and 5 is 3 5 = 2 = 2. The distance between 4 and 12 is 4 12 = 16 = 16. We saw an illustration in the first two examples above that a b = b a That makes sense. It means that the distance between a and b is the same as the distance between b and a, as it should be. We can check this fact using algebra: a b = (b a) = b a. As was discussed above, x 2 = x 2. Thus, substituting a b for x, we have a b 2 = (a b) 2 Lengths in right triangles A triangle is three points in the plane, with each pair of points joined by the straight line segment between them. A triangle is a figure with three sides, or a trigon. We usually place so much emphasis on the angles of three sided figures that we usually call them triangles instead of trigons. A right triangle is a triangle one of whose angles is a right angle. We ll have more to say about right angles and angles in general soon. Most people are comfortable with what a right angle is, although there are different names 179

3 for them. The three most common are an angle of 90, an angle of π 2, or one of four equal angles resulting from the intersecting of perpendicular lines. The side lengths of right triangles satisfy a well known equation. It s famous enough and useful enough that instead of calling it a claim we call it a theorem. The Pythagorean Theorem (1). If a, b, and c are the three lengths of the sides of a right triangle, and if c is the length of the side opposite the right angle in the triangle, then a 2 + b 2 = c 2 C. b 0 Proof: We can draw the same triangle four times to create a big square, each of whose sides have length a+b. This picture is drawn on the next page. 180

4 a. 6 6 b Notice that the area of the big square is b (a + b) 2 The big square can be dissected into2 4 triangles and a smaller square. Each of the 4 triangles has a base of a and a height of b, so they each have area 1 2ab. The smaller square in the middle of the picture has sides whose lengths equal c. Thus, the area of the square in the middle of the picture is c 2. The area of the entire picture is the sum of the areas of the 4 triangles and the smaller square in the middle 4( 1 ab) + c2 2 We ve calculated the area of the entire picture in two different ways, and they must be equal (a + b) 2 = 4( 1 ab) + c2 2 We can multiply out the left side of the equation and simplify the right side: Now subtract 2ab from both sides: a 2 + 2ab + b 2 = 2ab + c 2 a 2 + b 2 = c 2 181

5 Distance in R 2 The Pythagorean Theorem allows us to determine the distance between any pair of points in the plane. Proposition (2). The distance between two points (x 1, y 1 ) and (x 2, y 2 ) is (x1 x 2 ) 2 + (y 1 y 2 ) 2 Proof: The distance between (x 1, y 1 ) and (x 2, y 2 ) is the length, c, of a side of a right triangle. Notice that c is a length, so c 0. (x) C The other two sides of the triangle have length x 1 x 2 and y 1 y 2 so the Pythagorean theorem tells us that the distance, c, between (x 1, y 1 ) and (x 2, y 2 ) satisfies the equation Therefore, either c 2 = x 1 x y 1 y 2 2 c = x 1 x y 1 y 2 2 or c = x 1 x y 1 y 2 2 Recall that c 0, so the only solution for c is a Ix-x1 c = x 1 x y 1 y 2 2 We discussed earlier in this chapter that a b 2 = (a b) 2 for any numbers a and b, so we can write that c, the distance between (x 1, y 1 ) and (x 2, y 2 ), equals (x1 x 2 ) 2 + (y 1 y 2 ) 2 182

6 Example. The distance between the points (2, 3) and ( 5, 8) is (2 ( 5))2 + ( 3 8) 2 = (7) 2 + ( 11) 2 = = 170 Norms of vectors The norm of a vector (a, b) is a number, usually written as (a, b), that is the distance between the point (a, b) and the point (0, 0). Thus, (a, b) equals (a 0) 2 + (b 0) 2 which simplifies as (a, b) = a 2 + b 2 If you think of a vector as an arrow, then its norm is the length of the arrow. Example. The norm of the vector ( 1, 3) is ( 1, 3) = ( 1) = = 10 (-1,3) 3 b 0. Norms via dot products Recall from the exercises in the chapter The Plane of Vectors that ( a (a, b) = a b) 2 + b 2 is sometimes called the dot product of the vectors (a, b) and (a, b). Thus, the norm of (a, b) is the square root of the dot product of the vectors (a, b) and (a, b). That is how some prefer to think about norms. 183

7 Exercises For #1-14, provide the value asked for. 1.) 4 2.) 0 3.) 4 4.) ) ) 10 ( 4) 7.) the distance between 7 and 4 8.) the distance between 5 and 9 9.) the distance between (8, 2) and (2, 5) 10.) the distance between (3, 4) and (1, 7) 11.) the distance between (10, 4) and ( 2, 4) 12.) the distance between ( 3, 7) and ( 1, 5) 13.) the distance between ( 9, 3) and (0, 0) 14.) ( 6) For #15-18, give the norms of the vectors. 15.) (1, 3) 16.) ( 4, 6) 17.) ( 1, 5) 18.) (7, 3) For #19-21, find the solutions of the equations in one variable. 19.) (2x 2 3x 2) 2 = 1 20.) (2 e x ) 2 = 9 21.) log e (x) + 27 = 2 184

8 For #22-27, use the Pythagorean Theorem to determine the value of x, the length of the specified side of the right triangle. 22.) 25.) ) 26.) I0 x 7 24.) 27.) x I0 x

11.1 Vectors in the plane

11.1 Vectors in the plane What is a vector? It is an object having direction and length. Geometric way to represent vectors It is represented by an arrow. The direction of the arrow is the direction of