10.1 Vectors. c Kun Wang. Math 150, Fall 2017

Transcription

1 10.1 Vectors Definition. A vector is a quantity that has both magnitude and direction. A vector is often represented graphically as an arrow where the direction is the direction of the arrow, and the magnitude is the length of the arrow. Vectors are denoted by boldface letters a or by a. The magnitude of a vector a is denoted by a or a. In R 2, 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. Graphically, the vector is represented by the arrow with the tail at the origin and the tip of the arrow at the point (a 1, a 2 ). 1. Graph the vectors 5, 1. Given the points A(x 1, y 1 ) and B(x 2, y 2 ), the vector a with initial point A and terminal point B (also written AB) is a =< x2 x 1, y 2 y 1 >. Remarks: (1) A vector with initial point at the origin is said to be in standard position. (2) The only vector with length 0 is the zero vector 0 =< 0, 0 >. This vector is also the only vector with no specific direction. (3) Two vectors are equal if they have the same magnitude and direction. So, it doesnt matter where the vector is as long as it has the same magnitude and direction (the same displacement). 2. Graph the vector that starts at P (2, 1) and stops at Q(4, 5). Graph the vector 2, 4. These vectors have the same direction and magnitude (length) so they are the same vector. 1

2 Standard Basis Vectors: There are two special unit vectors that we use all the time: i =< 1, 0 >, j =< 0, 1 >. Then i and j are unit vectors pointing in the directions of the positive x- and y-axes. The vectors i and j are called coordinate vectors, sometimes they are denoted by e 1 and e 2. EVERY vector a =< a 1, a 2 > can be written in terms of these two vectors by a = a 1 i + a 2 j. Definition. In R 3, a two-dimensional vector is an ordered pair a = a 1, a 2, a 3 of real numbers. The numbers a 1, a 2 and a 3 are called the components of a. Graphically, the vector is represented by the arrow with the tail at the origin and the tip of the arrow at the point (a 1, a 2, a 3 ). Definition. In R 3 (three dimensional space), there are three special vectors, the vectors of length one in the positive direction of the x-, y-, and z-axis. These are sometimes called the coordinate vectors. They are denoted by the symbols i, j and k respectively, or e 1, e 2 and e 3 such that i = e 1 = 1, 0, 0 j = e 2 = 0, 1, 0 k = e 3 = 0, 0, 1 2

3 Change of Position Vectors can be used to show a change of position. 3. Find the vector which represents moving from the point P(-5, 2) to the point Q(3, -3). 4. Find the vector which represents moving from the point P(-4, 2, 1) to the point Q(5, 3, -4). Sketch the vector. 5. A dog runs 560 feet in a direction 43 west of north. Assuming that the origin is the dogs starting point and north is the positive y-axis, what are the coordinates of the dogs location? 6. In the previous problem suppose a flea was on the dogs back. If after the dog stops, the flea gets off and goes 1 foot northeast, what are the coordinates of the fleas location? 7. If a = 4, 7, then what angle between 0 and 360 does a make with the positive x-axis? 3

4 10.2 Scalar Multiplication Scalar Multiplication: The operation of scalar multiplication takes a scalar, that is a real number, and multiplies a vector by it. In R 2 (the plane), we define scalar multiplication of a scalar c and a vector a =< a 1, a 2 > as c a =< ca 1, ca 2 >. 1. For the scalar multiplication of 2 times the vector 1, 2 we get Note. 1. When we multiply a vector by a scalar, if the scalar is positive, the resulting vector has the same direction as the original. If the scalar is negative, the resulting vector has the opposite direction as the original vector. 2. When we multiply a vector by a scalar, the length of the new vector is the length of the old vector times the absolute value of the scalar, i.e., multiplying by the scalar 2 doubles the length, but multiplying by the scalar 1/2 cuts the length in half. For example, If you multiply a vector by a negative scalar such as 5, then the new vector is 5 times longer and points in the opposite direction of the original vector. 2. Find the scalar multiple of 5, 6 by the scalar a = 4. Sketch the original vector and resulting vector. 3. Does the equation a 5, 2 = 15, 6 have a solution? If it does, then find it. Definition. For vectors in R n (i.e., a vector in n-dimensions with n entries), we define scalar multiplication as c a =< a 1, a 2, a 3,, a n >=< ca 1, ca 2, ca 3,, ca n >. 4. Find the scalar multiple of 3, 7, 2 by the scalar -1/3. 4

5 10.3 Vector Addition and Subtraction Vector Addition: If a =< a 1, a 2 > and b =< b 1, b 2 >, then the vector a + b is defined by a + b =< a1 + b 1, a 2 + b 2 >. If a =< a 1, a 2,, a n > and b =< b 1, b 2,, b n >, then the vector a + b is defined by a + b =< a1 + b 1, a 2 + b 2, a n + b n >. Vector Difference: If a =< a 1, a 2 > and b =< b 1, b 2 >, then the vector a b is defined by a b =< a1 b 1, a 2 b 2 >. Vector Difference: If a =< a 1, a 2,, a n > and b =< b 1, b 2,, b n >, then the vector a b is defined by a b =< a1 b 1, a 2 b 2, a n b n >. Note. 1. This rule of addition is sometimes called the parallelogram law of addition. Notice in the graph above that the sum of the two vectors is the diagonal of the parallelogram formed by the two vectors. 2. If we plot v with its tail at the origin, and w with its tail at the tip of v, then the vector v + w is the vector stretching from the origin to the tip of the vectors w. 3. If we plot both v and w starting at the origin, then v w is the vector stretching from the tip of w to the tip of v. The vector w v is the vector stretching from the tip of v to the tip of w. 4. We can only add or subtract vectors if they have the same dimension (number of components). 5

6 5. Compute the following vectors and write the results in terms of the vectors i, j and k (a) 3 4, 2, 7 4 8, 5, 3 (b) 5, 6 5 2, 4 2 2, 2 6

7 10.4 Vector Length Definition. The length of any vector x 1, x 2 in R 2 is defined to be the distance form the origin to the point (x 1, x 2 ) which is x 1, x 2 = x x2 2 We use the notation x 1, x 2 to mean the length of the vector. A similar formula for the length of a vector exists for vectors in R n. x 1, x 2, x n = x x x2 n In particular, the length of the vector AB from A(x1, y 1 ) to B(x 2, y 2 ) is AB = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. Unit Vector: A vector with length (magnitude) 1 is called a Unit Vector. A unit vector in the direction of a is: 1. Given the points A( 2, 5) and B(4, 1), find the vector AB and its magnitude. Properties: Let X and Y be arbitrary vectors and let α and β be arbitrary scalars. Then we have the following properties. 1. X if and only if X is the zero vector (the vector in which all the components are zero. 2. α X = α X. That is the length of a scalar times a vector is the same as the length of the vector times the absolute value of the scalar. 3. X + Y X + Y. This is called the triangle inequality because the vectors X, Y, and X + Y can be thought of as the three sides of a triangle, and the sum of two side of the triangle must be longer than or equal to the third side. 2. Find the unit vector in the direction of

8 3. Find a vector of length 6, that is parallel to the vector which points from the point P (2, 5) to Q( 3, 8). 4. Consider a plane which is flying at 35,000 feet due north with a speed of 450 miles per hour. Suppose our plane suddenly experiences a down draft whose velocity is 125 mph. Find a vector to represent the planes velocity and find the speed of the plane. 5. Suppose a ferry is crossing a river 12 miles wide with a downstream current of 4 miles per hour. The ferry goes in a direction perpendicular to the bank at 10 miles per hour. How far downstream will the ferry reach the other bank? Also, find the velocity vector and the speed. 8

9 10.5 Dot Product The dot product of two nonzero vectors a and b, denoted a b, is the number a b = a b cos θ, where θ is the angle between a and b, 0 θ π. If the vectors are given in component form where a =< a 1, a 2 > and b =< b 1, b 2 >, then a b = a 1 b 1 + a 2 b 2. If a =< a 1, a 2,, a n > and b =< b 1, b 2,, b n >, then a b = a 1 b 1 + a 2 b a n b n. Important: The dot product of two vectors is always a SCALAR, not a vector. For this reason, the dot product is sometimes called the scalar product. 1. Calculate a b in the following scenarios. (1) a =< 4, 7 >, b =< 3, 3 > (2) a = 7, b = 2, and the angle between the two vectors is π/6. 2. Find the angle between the two vectors v = 4, 3 and w = 2, Find the angle between the two vectors v = 2, 1, 5 and w = 3, 2, 1 9

10 Properties of the Dot Product Theorem. The dot product of two vectors has the following properties. X and Y are vectors of the same dimension, and X = x 1, x 2,, x n. 1. X X = x x x2 n = X X Y = Y X 3. ( Y ) X + Z = X X + X Z. 4. a X Y = X a Y = a X Y. Perpendicular Vectors Two vectors a and b are parallel if b = c a for some scalar c. Two vectors are orthogonal or perpendicular if the angle between them is π/2 or 90. Thus, two vectors are orthogonal if and only if a b = 0, since cos π/2 = Determine if the vectors 3, 5, 4 and 12, 0, 9 are perpendicular. Orthogonal Complement: Sometimes it is useful to find a vector that is orthogonal to a given vector with the same length. Given the nonzero vector a =< a 1, a 2 >, the orthogonal complement of a is the vector a =< a 2, a 1 > 5. Find a unit vector that is perpendicular to the vector from P(-3, -4) to Q(4, -1). 6. A man walks 100 yards in a direction which is perpendicular to the vector 2, 13 and for which the x-value is decreasing. Find a vector of length 100 which points in this direction. 10