7.1 Vectors in 2-Space

INTRODUCTION

In science, mathematics, and engineering, we distinguish two important quantities: scalars and vectors. A scalar is simply a real number or a quantity that has magnitude. For example, length, temperature, and blood pressure are represented by numbers such as 80 m, 20°C, and the systolic/diastolic ratio 120/80, respectively. A vector, on the other hand, is usually described as a quantity that has both magnitude and direction.

Geometric Vectors

Geometrically, a vector can be represented by a directed line segment—that is, by an arrow—and is denoted either by a boldface symbol or by a symbol with an arrow over it; for example, v, , or . Examples of vector quantities shown in FIGURE 7.1.1 are weight w, velocity v, and the retarding force of friction Ff.

3 diagrams provide examples of vectors. A. A person lifts a barbell. 2 downward arrows from the weights at either end are labeled w. B. A person kicks a football. The ball moves along an arc. An arrow labeled v is tangent to the arc and points up and to the right. C. A person pushes a chest of drawers along the floor. A horizontal arrow labeled F subscript lowercase f points in the opposite direction of the movement.

FIGURE 7.1.1 Examples of vector quantities

Notation and Terminology

A vector whose initial point (or end) is A and whose terminal point (or tip) is B is written . The magnitude of a vector is written . Two vectors that have the same magnitude and same direction are said to be equal. Thus, in FIGURE 7.1.2, we have = . Vectors are said to be free, which means that a vector can be moved from one position to another provided its magnitude and direction are not changed. The negative of a vector , written − , is a vector that has the same magnitude as but is opposite in direction. If k ≠ 0 is a scalar, the scalar multiple of a vector, , is a vector that is |k| times as long as . If k > 0, then has the same direction as the vector ; if k < 0, then has the direction opposite that of . When k = 0, we say is the zero vector. Two vectors are parallel if and only if they are nonzero scalar multiples of each other. See FIGURE 7.1.3.

A vector labeled A B measures 3 units and goes up and to the left. A parallel vector labeled C D also measures 3 units and points up and to the left. The vectors are equal.

FIGURE 7.1.2 Vectors are equal

4 parallel vectors are graphed. Vector A B points up and to the right. Vector negative A B points down and to the left. Vector 3 over 2 times A B points up and to the right. It is 1.5 times longer than A B. Vector negative 1 over 4 times A B points down and to the left. It is one fourth the length of A B.

FIGURE 7.1.3 Parallel vectors

The question of what is the direction of 0 is usually answered by saying that the zero vector can be assigned any direction. More to the point, 0 is needed in order to have a vector algebra.

Addition and Subtraction

Two vectors can be considered as having a common initial point, such as A in FIGURE 7.1.4(a). Thus, if nonparallel vectors and are the sides of a parallelogram as in Figure 7.1.4(b), we say the vector that is the main diagonal, or , is the sum of and . We write

= + .

The difference of two vectors and is defined by

− = + (−).

2 graphs represent the sum of 2 vectors. Graph a. Two vectors A B and A C have a common initial point. Graph b. Vectors A B, A C, the parallel vector to A C from B, and the parallel vector to A B from C form a parallelogram A B C D. Vector A D is the sum of vectors A B and A C.

FIGURE 7.1.4 Vector is the sum of and

As seen in FIGURE 7.1.5(a), the difference − can be interpreted as the main diagonal of the parallelogram with sides and − . However, as shown in Figure 7.1.5(b), we can also interpret the same vector difference as the third side of a triangle with sides and . In this second interpretation, observe that the vector difference = − points toward the terminal point of the vector from which we are subtracting the second vector. If = , then

− = 0.

2 graphs represent the difference of 2 vectors. Graph a. Two vectors A B and A C have a common initial point. The vector negative A C is graphed from point A. The vertex of the parallelogram formed by sum of vectors A B and negative A C is marked. The vector from A to this point is the difference of vectors A B and A C. Graph b. Two vectors A B and A C have a common initial point. The vector joining C to B is the difference of vectors A B and A C.

FIGURE 7.1.5 Vector is the difference of and

Vectors in a Coordinate Plane

To describe a vector analytically, let us suppose for the remainder of this section that the vectors we are considering lie in a two-dimensional coordinate plane or 2-space. We shall denote the set of all vectors in the plane by R2. The vector shown in FIGURE 7.1.6, with initial point the origin O and terminal point P(x1, y1), is called the position vector of the point P and is written

Vector O P begins at the origin O and goes up and to the right to a point P. The coordinates of P are (x subscript 1, y subscript 1).

FIGURE 7.1.6 Position vector

Components

In general, a vector a in R2 is any ordered pair of real numbers,

The numbers a1 and a2 are said to be the components of the vector a.

As we shall see in the first example, the vector a is not necessarily a position vector.

EXAMPLE 1 Position Vector

The displacement from the initial point P1(x, y) to the terminal point P2(x + 4, y + 3) in FIGURE 7.1.7(a) is 4 units to the right and 3 units up. As seen in Figure 7.1.7(b), the position vector of a = ⟨4, 3〉 emanating from the origin is equivalent to the displacement vector from P1(x, y) to P2(x + 4, y + 3). ≡

2 graphs represent the displacement of a vector P subscript 1 P subscript 2 to the origin. Graph a. A vector P subscript 1 P subscript 2 goes up and to the right from point P subscript 1 (x y) to point P subscript 2 (x + 4, y + 3). Graph b. The vector is displaced to begin at the origin. The length of the vector is a and it ends at point P (4, 3).

FIGURE 7.1.7 Graphs of vectors in Example 1

Addition and subtraction of vectors, multiplication of vectors by scalars, and so on, are defined in terms of their components.

DEFINITION 7.1.1 Addition, Scalar Multiplication, Equality

Let a = 〈a1, a2〉 and b = 〈b1, b2〉 be vectors in R2.

  1. Addition: a + b = 〈a1 + b1, a2 + b2〉 (1)
  2. Scalar multiplication: ka = 〈ka1, ka2〉 (2)
  3. Equality: a = b if and only if a1 = b1, a2 = b2 (3)

Subtraction

Using (2), we define the negative of a vector b by

−b = (−1)b = 〈−b1, −b2〉.

We can then define the subtraction, or the difference, of two vectors as

a − b = a + (−b) = 〈a1 − b1, a2 − b2〉. (4)

In FIGURE 7.1.8(a) on page 331, we have illustrated the sum of two vectors and . In Figure 7.1.8(b), the vector , with initial point P1 and terminal point P2, is the difference of position vectors

= − = 〈x2 − x1, y2 − y1〉.

The sum and the difference of 2 vectors are graphed as follows. Graph a. 2 vectors O P subscript 1 and O P subscript 2 are position vectors in quadrant 1. The coordinates of the end points are P subscript 1 (x subscript 1, y subscript 1) and P subscript 2 (x subscript 2, y subscript 2). Vectors O P subscript 1, O P subscript 2, the parallel vector to O P subscript 2 from P subscript 1, and the parallel vector to O P subscript 1 from P subscript 2 form a parallelogram. The diagonal vector O P is the sum of vectors O P subscript 1 and O P subscript 2. The coordinates of P are (x subscript 1 + x subscript 2, y subscript 1 + y subscript 2). Graph b. 2 vectors O P subscript 1 and O P subscript 2 are position vectors in quadrant 1. The coordinates of the end points are P subscript 1 (x subscript 1, y subscript 1) and P subscript 2 (x subscript 2, y subscript 2). The vector joining P subscript 1 to P subscript 2 is the difference P sub 1 P sub 2 of vectors O P sub 1 and O P sub 2. When displaced to the origin, the coordinates of the vector are O P (x subscript 2 minus x subscript 1, y subscript 2 minus y subscript 1).

FIGURE 7.1.8 In (b), and are the same vector

As shown in Figure 7.1.8(b), the vector can be drawn either starting from the terminal point of and ending at the terminal point of , or as the position vector whose terminal point has coordinates (x2 − x1, y2 − y1). Remember, and are considered equal, since they have the same magnitude and direction.

EXAMPLE 2 Addition and Subtraction of Two Vectors

If a = 〈1, 4〉 and b = 〈−6, 3〉, find a + b, a − b, and 2a + 3b.

SOLUTION

We use (1), (2), and (4):

≡

Properties

The component definition of a vector can be used to verify each of the following properties of vectors in R2:

THEOREM 7.1.1 Properties of Vectors

  1. a + b = b + a ← commutative law
  2. a + (b + c) = (a + b) + c ← associative law
  3. a + 0 = a ← additive identity
  4. a + (−a) = 0 ← additive inverse
  5. k(a + b) = ka + kb, k a scalar
  6. (k1 + k2)a = k1a + k2a, k1 and k2 scalars
  7. k1(k2a) = (k1k2)a, k1 and k2 scalars
  8. 1a = a
  9. 0a = 0 ← zero vector

The zero vector 0 in properties (iii), (iv), and (ix) is defined as

0 = 〈0, 0〉.

Magnitude

The magnitude, length, or norm of a vector a is denoted by . Motivated by the Pythagorean theorem and FIGURE 7.1.9, we define the magnitude of a vector

Clearly, ≥ 0 for any vector a, and = 0 if and only if a = 0. For example, if a = 〈6, −2〉, then =

A position vector measures a units. It goes up and to the right from the origin into the first quadrant. A vertical dotted line from the end point to the x axis is labeled a subscript 2. The distance from the origin to the dotted line is labeled a subscript 1. a, a subscript 1, and a subscript 2 form a right triangle.

FIGURE 7.1.9 A right triangle

Unit Vectors

A vector that has magnitude 1 is called a unit vector. We can obtain a unit vector u in the same direction as a nonzero vector a by multiplying a by the positive scalar k = (reciprocal of its magnitude). In this case we say that is the normalization of the vector a. The normalization of the vector a is a unit vector because

Note: It is often convenient to write the scalar multiple u = as

.

EXAMPLE 3 Unit Vectors

Given a = 〈2, −1〉, form a unit vector in the same direction as a. In the opposite direction of a.

SOLUTION

The magnitude of the vector a is = . Thus a unit vector in the same direction as a is the scalar multiple

A unit vector in the opposite direction of a is the negative of u:

≡

If a and b are vectors and c1 and c2 are scalars, then the expression c1a + c2b is called a linear combination of a and b. As we see next, any vector in R2 can be written as a linear combination of two special vectors.

The i, j Vectors

In view of (1) and (2), any vector a = 〈a1, a2〉 can be written as a sum:

〈a1, a2〉 = 〈a1, 0〉 + 〈0, a2〉 = a1〈1, 0〉 + a2〈0, 1〉. (5)

The unit vectors 〈1, 0〉 and 〈0, 1〉 are usually given the special symbols i and j. See FIGURE 7.1.10(a). Thus, if

i = 〈1, 0〉 and j = 〈0, 1〉,

then (5) becomes a = a1i + a2j. (6)

Graph a. Unit vectors i and j are graphed on the horizontal and vertical axes respectively. Graph b. A position vector measures a units. A vertical dotted line from the end point intersects the x axis at a point. The vector joining the origin to the point is labeled a subscript 1 i. A horizontal dotted line from the end point of the position vector intersects the y axis at a point. The vector joining the origin to the point is labeled a subscript 2 j.

FIGURE 7.1.10 i and j form a basis for R2

The unit vectors i and j are referred to as the standard basis for the system of two-dimensional vectors, since any vector a can be written uniquely as a linear combination of i and j. If a = a1i + a2j is a position vector, then Figure 7.1.10(b) shows that a is the sum of the vectors a1i and a2j, which have the origin as a common initial point and which lie on the x- and y-axes, respectively. The scalar is called the horizontal component of a, and the scalar is called the vertical component of a.

EXAMPLE 4 Vector Operations Using i and j

  1. 〈4, 7〉 = 4i + 7j
  2. (2i − 5j) + (8i + 13j) = 10i + 8j
  3. =
  4. 10(3i − j) = 30i − 10j
  5. a = 6i + 4j and b = 9i + 6j are parallel, since b is a scalar multiple of a. We see that b = a. ≡

EXAMPLE 5 Graphs of Vector Sum/Vector Difference

Let a = 4i + 2j and b = −2i + 5j. Graph a + b and a − b.

SOLUTION

The graphs of a + b = 2i + 7j and a − b = 6i − 3j are given in FIGURE 7.1.11(a) and 7.1.11(b), respectively.

The sum and the difference of 2 vectors are graphed as follows. Graph a. Vectors a and b are position vectors. Vector a goes 4 units to the right and 2 units up. Vector b goes 2 units to the left and 5 units up. The diagonal a + b of the parallelogram formed by sides a and b goes 2 units to the right and 7 units up. Graph b. Vectors a and b are position vectors. Vector a goes 4 units to the right and 2 units up. Vector b goes 2 units to the left and 5 units up. The vector joining b to a is the difference a minus b. When displaced to the origin, it goes 6 units to the right and 3 units down.

FIGURE 7.1.11 Graphs of vectors in Example 5 ≡

7.1 Exercises Answers to selected odd-numbered problems begin on page ANS-15.

In Problems 1–8, find (a) 3a, (b) a + b, (c) a − b, (d) , and (e) .

  1. a = 2i + 4j, b = −i + 4j
  2. a = 〈1, 1〉, b = 〈2, 3〉
  3. a = 〈4, 0〉, b = 〈0, −5〉
  4. a = i − j, b = i + j
  5. a = −3i + 2j, b = 7j
  6. a = 〈1, 3〉, b = −5a
  7. a = −b, b = 2i − 9j
  8. a = 〈7, 10〉, b = 〈1, 2〉

In Problems 9–14, find (a) 4a − 2b and (b) −3a − 5b.

  1. a = 〈1, −3〉, b = 〈−1, 1〉
  2. a = i + j, b = 3i − 2j
  3. a = i − j, b = −3i + 4j
  4. a = 〈2, 0〉, b = 〈0, −3〉
  5. a = 〈4, 10〉, b = −2〈1, 3〉
  6. a = 〈3, 1〉 + 〈−1, 2〉, b = 〈6, 5〉 − 〈1, 2〉

In Problems 15–18, find the vector . Graph and its corresponding position vector.

  1. P1(3, 2), P2(5, 7)
  2. P1(−2, −1), P2(4, −5)
  3. P1(3, 3), P2(5, 5)
  4. P1(0, 3), P2(2, 0)
  5. Find the terminal point of the vector = 4i + 8j if its initial point is (−3, 10).
  6. Find the initial point of the vector = 〈−5, −1〉 if its terminal point is 〈4, 7〉.
  7. Determine which of the following vectors are parallel to a = 4i + 6j.
    1. −4i − 6j
    2. −i − j
    3. 10i + 15j
    4. 2(i − j) − 3(i − j)
    5. 8i + 12j
    6. (5i + j) − (7i + 4j)
  8. Determine a scalar c so that a = 3i + cj and b = −i + 9j are parallel.

In Problems 23 and 24, find a + (b + c) for the given vectors.

  1. a = 〈5, 1〉, b = 〈−2, 4〉, c = 〈3, 10〉
  2. a = 〈1, 1〉, b = 〈4, 3〉, c = 〈0, −2〉

In Problems 25–28, find a unit vector (a) in the same direction as a, and (b) in the opposite direction of a.

  1. a = 〈2, 2〉
  2. a = 〈−3, 4〉
  3. a = 〈0, −5〉
  4. a = 〈1, −〉

In Problems 29 and 30, a = 〈2, 8〉 and b = 〈3, 4〉. Find a unit vector in the same direction as the given vector.

  1. a + b
  2. 2a − 3b

In Problems 31 and 32, find a vector b that is parallel to the given vector and has the indicated magnitude.

  1. a = 3i + 7j, = 2
  2. a = i − j, = 3
  3. Find a vector in the opposite direction of a = 〈4, 10〉 but as long.
  4. Given that a = 〈1, 1〉 and b = 〈−1, 0〉, find a vector in the same direction as a + b but five times as long.

In Problems 35 and 36, use the given figure to draw the indicated vector.

  1. 3b − a
    2 vectors a and b have a common initial point. Vector a points up and to the left. Vector b measures half the length of vector a and points up and to the right.

    FIGURE 7.1.12 Vectors for Problem 35

  2. a + (b + c)
    3 vectors a, b, and c have a common initial point and are of equal length. Vector a points to the left, vector b points up, and vector c points to the right.

    FIGURE 7.1.13 Vectors for Problem 36

In Problems 37 and 38, express the vector x in terms of the vectors a and b.

  1. Vectors a and b have a common initial point. Vector a goes up and to the right, vector b goes to the right. A parallelogram with a and b as sides is drawn. Vector x is the diagonal from the vertex opposite the initial point to the initial point.

    FIGURE 7.1.14 Vector x in Problem 37

  2. Vectors a and b have a common initial point. Vector a goes up and to the right, vector b goes to the right. Vector x begins at the endpoint of vector b and goes up and to the left. The endpoint of vector a is the midpoint of x.

    FIGURE 7.1.15 Vector x in Problem 38

In Problems 39 and 40, use the given figure to prove the given result.

  1. a + b + c = 0
    3 vectors a, b, and c form a triangle. Vector a points to the right, vector b points up and to the left, and vector c points down and to the left.

    FIGURE 7.1.16 Vectors for Problem 39

  2. a + b + c + d = 0
    Vectors a, b, and c form a quadrilateral. Vector a points to the right, vector b points up and to the right, vector c points up and to the left, and vector d points down and to the left.

    FIGURE 7.1.17 Vectors for Problem 40

In Problems 41 and 42, express the vector a = 2i + 3j as a linear combination of the given vectors b and c.

  1. b = i + j, c = i − j
  2. b = −2i + 4j, c = 5i + 7j

A vector is said to be tangent to a curve at a point if it is parallel to the tangent line at the point. In Problems 43 and 44, find a unit tangent vector to the given curve at the indicated point.

  1. y = x2 + 1, (2, 2)
  2. y = −x2 + 3x, (0, 0)
  3. When walking, a person’s foot strikes the ground with a force F at an angle θ from the vertical. In FIGURE 7.1.18, the vector F is resolved into vector components Fg, which is parallel to the ground, and Fn, which is perpendicular to the ground. In order that the foot does not slip, the force Fg must be offset by the opposing force Ff of friction; that is, Ff = −Fg.
    1. Use the fact that = µ, where µ is the co-efficient of friction, to show that tan θ = µ. The foot will not slip for angles less than or equal to θ.
    2. Given that µ = 0.6 for a rubber heel striking an asphalt sidewalk, find the “no-slip” angle.
    A person’s foot strikes the ground with a force F at an angle theta from the vertical. The horizontal and vertical components of force F are F subscript g and F subscript n. Force F subscript lowercase f is equal to F subscript g but in the opposite direction.

    FIGURE 7.1.18 Vector F in Problem 45

  1. A 200-lb traffic light supported by two cables hangs in equilibrium. As shown in FIGURE 7.1.19(b), let the weight of the light be represented by w and the forces in the two cables by F1 and F2. From Figure 7.1.19(c), we see that a condition of equilibrium is

    w + F1 + F2 = 0. (7)

    See Problem 39. If

    use (7) to determine the magnitudes of F1 and F2. [Hint: Reread (iii) of Definition 7.1.1.]

    3 figures represent the forces acting on a traffic light. Graph a. 2 cables support a traffic light. The cables form angles of 15 degrees and 20 degrees above the horizontal. Graph b. There are 3 forces acting on the traffic signal. F subscript 1 is along the cable that is 20 degrees above horizontal, F subscript 2 is along the cable that is 15 degrees above horizontal, and w is downward. Graph c. The 3 forces F subscript 1, F subscript 2, and w form a triangle.

    FIGURE 7.1.19 Three force vectors in Problem 46

  2. An electric charge Q is uniformly distributed along the y-axis between y = −a and y = a. See FIGURE 7.1.20. The total force exerted on the charge q on the x-axis by the charge Q is F = Fxi + Fyj, where

    Determine F.
    The diagram represents a charge Q ranging from negative a to a on the y-axis, and a charge lowercase q at a distance L on the positive x-axis.

    FIGURE 7.1.20 Charge on x-axis in Problem 47

  3. Using vectors, show that the diagonals of a parallelogram bisect each other. [Hint: Let M be the midpoint of one diagonal and N the midpoint of the other.]
  4. Using vectors, show that the line segment between the midpoints of two sides of a triangle is parallel to the third side and half as long.
  5. An airplane starts from an airport located at the origin O and flies 150 miles in the direction 20° north of east to city A. From A, the airplane then flies 200 miles in the direction 23° west of north to city B. From B, the airplane flies 240 miles in the direction 10° south of west to city C. Express the location of city C as a vector r as shown in FIGURE 7.1.21. Find the distance from O to C.
    An airplane flies from the origin O to 3 points A, B, and C. Vectors represent the trajectory on a coordinate plane. A compass indicates north as up. A vector goes up and to the right 20 degrees above horizontal from O to A. Another vector goes up and to the left from A to B at an angle 23 degrees left of the vertical. A third vector from B to C goes down and to the left 10 degrees below horizontal. A fourth vector is labeled r and goes up and to the left from O to C.

    FIGURE 7.1.21 Airplane in Problem 50