Vectors in the Plane

When describing the movement of an airplane in flight, it is important to communicate two pieces of information: the direction in which the plane is traveling and the plane’s speed. When measuring a force, such as the thrust of the plane’s engines, it is important to describe not only the strength of that force, but also the direction in which it is applied. Some quantities, such as or force, are defined in terms of both size (also called magnitude) and direction. A quantity that has magnitude and direction is called a vector. In this text, we denote vectors by boldface letters, such as v.

Vector Representation

A vector in a plane is represented by a directed line segment (an arrow). The endpoints of the segment are called the initial point and the terminal point of the vector. An arrow from the initial point to the terminal point indicates the direction of the vector. The length of the line segment represents its magnitude. We use the notation

‖ v ‖

A vector with an initial point and terminal point that are the same is called the zero vector, denoted

0 .

The zero vector is the only vector without a direction, and by convention can be considered to have any direction convenient to the problem at hand.

Vectors with the same magnitude and direction are called equivalent vectors. We treat equivalent vectors as equal, even if they have different initial points. Thus, if

v

The arrows in [link](b) are equivalent. Each arrow has the same length and direction. A closely related concept is the idea of parallel vectors. Two vectors are said to be parallel if they have the same or opposite directions. We explore this idea in more detail later in the chapter. A vector is defined by its magnitude and direction, regardless of where its initial point is located.

The use of boldface, lowercase letters to name vectors is a common representation in print, but there are alternative notations. When writing the name of a vector by hand, for example, it is easier to sketch an arrow over the variable than to simulate boldface type:

\vec{v} .

Combining Vectors

Vectors have many real-life applications, including situations involving force or velocity. For example, consider the forces acting on a boat crossing a river. The boat’s motor generates a force in one direction, and the current of the river generates a force in another direction. Both forces are vectors. We must take both the magnitude and direction of each force into account if we want to know where the boat will go.

A second example that involves vectors is a quarterback throwing a football. The quarterback does not throw the ball parallel to the ground; instead, he aims up into the air. The velocity of his throw can be represented by a vector. If we know how hard he throws the ball (magnitude—in this case, speed), and the angle (direction), we can tell how far the ball will travel down the field.

A real number is often called a scalar in mathematics and physics. Unlike vectors, scalars are generally considered to have a magnitude only, but no direction. Multiplying a vector by a scalar changes the vector’s magnitude. This is called scalar multiplication. Note that changing the magnitude of a vector does not indicate a change in its direction. For example, wind blowing from north to south might increase or decrease in speed while maintaining its direction from north to south.

Another operation we can perform on vectors is to add them together in vector addition, but because each vector may have its own direction, the process is different from adding two numbers. The most common graphical method for adding two vectors is to place the initial point of the second vector at the terminal point of the first, as in [link](a). To see why this makes sense, suppose, for example, that both vectors represent displacement. If an object moves first from the initial point to the terminal point of vector

v,

the overall displacement is the same as if the object had made just one movement from the initial point to the terminal point of the vector

v + w .

For obvious reasons, this approach is called the triangle method. Notice that if we had switched the order, so that

w

was our first vector and v was our second vector, we would have ended up in the same place. (Again, see [link](a).) Thus,

v + w = w + v .

A second method for adding vectors is called the parallelogram method. With this method, we place the two vectors so they have the same initial point, and then we draw a parallelogram with the vectors as two adjacent sides, as in [link](b). The length of the diagonal of the parallelogram is the sum. Comparing [link](b) and [link](a), we can see that we get the same answer using either method. The vector

v + w

These three vectors form the sides of a triangle. It follows that the length of any one side is less than the sum of the lengths of the remaining sides. So we have

This is known more generally as the triangle inequality. There is one case, however, when the resultant vector

u + v

Vector Components

Working with vectors in a plane is easier when we are working in a coordinate system. When the initial points and terminal points of vectors are given in Cartesian coordinates, computations become straightforward.

We have seen how to plot a vector when we are given an initial point and a terminal point. However, because a vector can be placed anywhere in a plane, it may be easier to perform calculations with a vector when its initial point coincides with the origin. We call a vector with its initial point at the origin a standard-position vector. Because the initial point of any vector in standard position is known to be

(0, 0),

we can describe the vector by looking at the coordinates of its terminal point. Thus, if vector v has its initial point at the origin and its terminal point at

(x, y),

When a vector is written in component form like this, the scalars x and y are called the components of

v .

Recall that vectors are named with lowercase letters in bold type or by drawing an arrow over their name. We have also learned that we can name a vector by its component form, with the coordinates of its terminal point in angle brackets. However, when writing the component form of a vector, it is important to distinguish between

〈 x, y 〉

The first ordered pair uses angle brackets to describe a vector, whereas the second uses parentheses to describe a point in a plane. The initial point of

〈 x, y 〉

When we have a vector not already in standard position, we can determine its component form in one of two ways. We can use a geometric approach, in which we sketch the vector in the coordinate plane, and then sketch an equivalent standard-position vector. Alternatively, we can find it algebraically, using the coordinates of the initial point and the terminal point. To find it algebraically, we subtract the x-coordinate of the initial point from the x-coordinate of the terminal point to get the x component, and we subtract the y-coordinate of the initial point from the y-coordinate of the terminal point to get the y component.

To find the magnitude of a vector, we calculate the distance between its initial point and its terminal point. The magnitude of vector

v = 〈 x, y 〉

Note that because this vector is written in component form, it is equivalent to a vector in standard position, with its initial point at the origin and terminal point

(x, y) .

Thus, it suffices to calculate the magnitude of the vector in standard position. Using the distance formula to calculate the distance between initial point

(0, 0)

The magnitude of a vector can also be derived using the Pythagorean theorem, as in the following figure.

We have defined scalar multiplication and vector addition geometrically. Expressing vectors in component form allows us to perform these same operations algebraically.

Now that we have established the basic rules of vector arithmetic, we can state the properties of vector operations. We will prove two of these properties. The others can be proved in a similar manner.

Proof of Commutative Property

Proof of Distributive Property

We have found the components of a vector given its initial and terminal points. In some cases, we may only have the magnitude and direction of a vector, not the points. For these vectors, we can identify the horizontal and vertical components using trigonometry ([link]).

formed by the vector v and the positive x-axis. We can see from the triangle that the components of vector

v

Therefore, given an angle and the magnitude of a vector, we can use the cosine and sine of the angle to find the components of the vector.

Unit Vectors

Recall that when we defined scalar multiplication, we noted that

‖ k v ‖ = \| k \| \cdot ‖ v ‖ .

([link]). The process of using scalar multiplication to find a unit vector with a given direction is called normalization.

We have seen how convenient it can be to write a vector in component form. Sometimes, though, it is more convenient to write a vector as a sum of a horizontal vector and a vertical vector. To make this easier, let’s look at standard unit vectors. The standard unit vectors are the vectors

i = 〈 1, 0 〉

By applying the properties of vectors, it is possible to express any vector in terms of

i

Applications of Vectors

Because vectors have both direction and magnitude, they are valuable tools for solving problems involving such applications as motion and force. Recall the boat example and the quarterback example we described earlier. Here we look at two other examples in detail.

Key Concepts

For the following exercises, consider points $P (−1, 3),$

Q (1, 5),

and $R (−3, 7) .$

Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.

\vec{P Q}

a. $\vec{P Q} = 〈 2, 2 〉;$

b. $\vec{P Q} = 2 i + 2 j$

\vec{P R}

\vec{Q P}

a. $\vec{Q P} = 〈 −2, −2 〉;$

b. $\vec{Q P} = −2 i - 2 j$

\vec{R P}

\vec{P Q} + \vec{P R}

a. $\vec{P Q} + \vec{P R} = 〈 0, 6 〉;$

b. $\vec{P Q} + \vec{P R} = 6 j$

\vec{P Q} - \vec{P R}

2 \vec{P Q} - 2 \vec{P R}

a. $2 \vec{P Q} - 2 \vec{P R} = 〈 8, −4 〉;$

b. $2 \vec{P Q} - 2 \vec{P R} = 8 i - 4 j$

2 \vec{P Q} + \frac{1}{2} \vec{P R}

The unit vector in the direction of $\vec{P Q}$

a. $〈 \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} 〉;$

b. $\frac{1}{\sqrt{2}} i + \frac{1}{\sqrt{2}} j$

The unit vector in the direction of $\vec{P R}$

A vector $v$

has initial point $(−1, −3)$

and terminal point $(2, 1) .$

Find the unit vector in the direction of $v .$

Express the answer in component form.

〈 \frac{3}{5}, \frac{4}{5} 〉

A vector $v$

has initial point $(−2, 5)$

and terminal point $(3, −1) .$

Find the unit vector in the direction of $v .$

Express the answer in component form.

The vector $v$

has initial point $P (1, 0)$

and terminal point $Q$

that is on the y-axis and above the initial point. Find the coordinates of terminal point $Q$

such that the magnitude of the vector $v$

is $\sqrt{5} .$

Q (0, 2)

The vector $v$

has initial point $P (1, 1)$

and terminal point $Q$

that is on the x-axis and left of the initial point. Find the coordinates of terminal point $Q$

such that the magnitude of the vector $v$

is $\sqrt{10} .$

For the following exercises, use the given vectors $a$

and $b .$

Determine the vector sum $a + b$
and express it in both the component form and by using the standard unit vectors.
Find the vector difference $a - b$
and express it in both the component form and by using the standard unit vectors.
Verify that the vectors $a,$ $b,$
and
$a + b,$
and, respectively,
$a,$ $b,$
and
$a - b$
satisfy the triangle inequality.
Determine the vectors $2 a,$ $− b,$
and
$2 a - b .$
Express the vectors in both the component form and by using standard unit vectors.

a = 2 i + j,

b = i + 3 j

a. $a + b = 3 i + 4 j,$

a + b = 〈 3, 4 〉;

b. $a - b = i - 2 j,$

a - b = 〈 1, −2 〉;

c. Answers will vary; d. $2 a = 4 i + 2 j,$

2 a = 〈 4, 2 〉,

− b = − i - 3 j,

− b = 〈 −1, −3 〉,

2 a - b = 3 i - j,

2 a - b = 〈 3, −1 〉

a = 2 i,

b = −2 i + 2 j

Let $a$

be a standard-position vector with terminal point $(−2, −4) .$

Let $b$

be a vector with initial point $(1, 2)$

and terminal point $(−1, 4) .$

Find the magnitude of vector $−3 a + b - 4 i + j .$

15

Let $a$

be a standard-position vector with terminal point at $(2, 5) .$

Let $b$

be a vector with initial point $(−1, 3)$

and terminal point $(1, 0) .$

Find the magnitude of vector $a - 3 b + 14 i - 14 j .$

Let $u$

and $v$

be two nonzero vectors that are nonequivalent. Consider the vectors $a = 4 u + 5 v$

and $b = u + 2 v$

defined in terms of $u$

and $v .$

Find the scalar $λ$

such that vectors $a + λ b$

and $u - v$

are equivalent.

λ = −3

Let $u$

and $v$

be two nonzero vectors that are nonequivalent. Consider the vectors $a = 2 u - 4 v$

and $b = 3 u - 7 v$

defined in terms of $u$

and $v .$

Find the scalars $α$

and $β$

such that vectors $α a + β b$

and $u - v$

are equivalent.

Consider the vector $a (t) = 〈 cos t, sin t 〉$

with components that depend on a real number $t .$

As the number $t$

varies, the components of $a (t)$

change as well, depending on the functions that define them.

Write the vectors $a (0)$
and
$a (π)$
in component form.
Show that the magnitude $‖ a (t) ‖$
of vector
$a (t)$
remains constant for any real number
$t .$
As $t$
varies, show that the terminal point of vector
$a (t)$
describes a circle centered at the origin of radius
$1 .$

a. $a (0) = 〈 1, 0 〉,$

a (π) = 〈 −1, 0 〉;

b. Answers may vary; c. Answers may vary

Consider vector $a (x) = 〈 x, \sqrt{1 - x^{2}} 〉$

with components that depend on a real number $x \in [−1, 1] .$

As the number $x$

varies, the components of $a (x)$

change as well, depending on the functions that define them.

Write the vectors $a (0)$
and
$a (1)$
in component form.
Show that the magnitude $‖ a (x) ‖$
of vector
$a (x)$
remains constant for any real number
$x$
As $x$
varies, show that the terminal point of vector
$a (x)$
describes a circle centered at the origin of radius
$1 .$

Show that vectors $a (t) = 〈 cos t, sin t 〉$

and $a (x) = 〈 x, \sqrt{1 - x^{2}} 〉$

are equivalent for $x = r$

and $t = 2 k π,$

where $k$

is an integer.

Answers may vary

Show that vectors $a (t) = 〈 cos t, sin t 〉$

and $a (x) = 〈 x, \sqrt{1 - x^{2}} 〉$

are opposite for $x = r$

and $t = π + 2 k π,$

where $k$

is an integer.

For the following exercises, find vector $v$

with the given magnitude and in the same direction as vector $u .$

‖ v ‖ = 7, u = 〈 3, 4 〉

v = 〈 \frac{21}{5}, \frac{28}{5} 〉

‖ v ‖ = 3, u = 〈 −2, 5 〉

‖ v ‖ = 7, u = 〈 3, −5 〉

v = 〈 \frac{21 \sqrt{34}}{34}, - \frac{35 \sqrt{34}}{34} 〉

‖ v ‖ = 10, u = 〈 2, −1 〉

For the following exercises, find the component form of vector $u,$

given its magnitude and the angle the vector makes with the positive x-axis. Give exact answers when possible.

‖ u ‖ = 2,

θ = 30 °

u = 〈 \sqrt{3}, 1 〉

‖ u ‖ = 6,

θ = 60 °

‖ u ‖ = 5,

θ = \frac{π}{2}

u = 〈 0, 5 〉

‖ u ‖ = 8,

θ = π

‖ u ‖ = 10,

θ = \frac{5 π}{6}

u = 〈 −5 \sqrt{3}, 5 〉

‖ u ‖ = 50,

θ = \frac{3 π}{4}

For the following exercises, vector $u$

is given. Find the angle $θ \in [0, 2 π)$

that vector $u$

makes with the positive direction of the x-axis, in a counter-clockwise direction.

u = 5 \sqrt{2} i - 5 \sqrt{2} j

θ = \frac{7 π}{4}

u = − \sqrt{3} i - j

Let $a = 〈 a_{1}, a_{2} 〉,$

b = 〈 b_{1}, b_{2} 〉,

and $c = 〈 c_{1}, c_{2} 〉$

be three nonzero vectors. If $a_{1} b_{2} - a_{2} b_{1} \neq 0,$

then show there are two scalars, $α$

and $β,$

such that $c = α a + β b .$

Answers may vary

Consider vectors $a = 〈 2, −4 〉,$

b = 〈 −1, 2 〉,

and c = 0 Determine the scalars $α$

and $β$

such that $c = α a + β b .$

Let $P (x_{0}, f (x_{0}))$

be a fixed point on the graph of the differential function $f$

with a domain that is the set of real numbers.

Determine the real number $z_{0}$
such that point
$Q (x_{0} + 1, z_{0})$
is situated on the line tangent to the graph of
$f$
at point
$P .$
Determine the unit vector $u$
with initial point
$P$
and terminal point
$Q .$

a. $z_{0} = f (x_{0}) + f^{'} (x_{0});$

b. $u = \frac{1}{\sqrt{1 + {[f^{'} (x_{0})]}^{2}}} 〈 1, f^{'} (x_{0}) 〉$

Consider the function $f (x) = x^{4},$

where $x \in ℝ .$

Determine the real number $z_{0}$
such that point
$Q (2, z_{0})$
s situated on the line tangent to the graph of
$f$
at point
$P (1, 1) .$
Determine the unit vector $u$
with initial point
$P$
and terminal point
$Q .$

Consider $f$

and $g$

two functions defined on the same set of real numbers $D .$

Let $a = 〈 x, f (x) 〉$

and $b = 〈 x, g (x) 〉$

be two vectors that describe the graphs of the functions, where $x \in D .$

Show that if the graphs of the functions $f$

and $g$

do not intersect, then the vectors $a$

and $b$

are not equivalent.

Find $x \in ℝ$

such that vectors $a = 〈 x, sin x 〉$

and $b = 〈 x, cos x 〉$

are equivalent.

Calculate the coordinates of point $D$

such that $A B C D$

is a parallelogram, with $A (1, 1),$

B (2, 4),

and $C (7, 4) .$

D (6, 1)

Consider the points $A (2, 1),$

B (10, 6),

C (13, 4),

and $D (16, −2) .$

Determine the component form of vector $\vec{A D} .$

The speed of an object is the magnitude of its related velocity vector. A football thrown by a quarterback has an initial speed of $70$

mph and an angle of elevation of $30 ° .$

Determine the velocity vector in mph and express it in component form. (Round to two decimal places.)

This figure shows a football being thrown. The path of the football is represented by an arcing curve. At the beginning of the curve is a vector indicating the initial velocity.

〈 60.62, 35 〉

A baseball player throws a baseball at an angle of $30 °$

with the horizontal. If the initial speed of the ball is $100$

mph, find the horizontal and vertical components of the initial velocity vector of the baseball. (Round to two decimal places.)

A bullet is fired with an initial velocity of $1500$

ft/sec at an angle of $60 °$

with the horizontal. Find the horizontal and vertical components of the velocity vector of the bullet. (Round to two decimal places.)

This figure is the first quadrant of a coordinate system. There is a vector from the origin that is labeled “v sub 0 = 1500 feet per second.” The angle between the x-axis and the vector is 60 degrees.

The horizontal and vertical components are $750$

ft/sec and $1299.04$

ft/sec, respectively.

[T] A 65-kg sprinter exerts a force of $798$

N at a $19 °$

angle with respect to the ground on the starting block at the instant a race begins. Find the horizontal component of the force. (Round to two decimal places.)

[T] Two forces, a horizontal force of $45$

lb and another of $52$

lb, act on the same object. The angle between these forces is $25 ° .$

Find the magnitude and direction angle from the positive x-axis of the resultant force that acts on the object. (Round to two decimal places.)

This figure has two vectors with the same initial point. The first vector is labeled “52 lb” and the second vector is labeled “45 lb.” The angle between the vectors is 25 degrees.

The magnitude of resultant force is $94.71$

lb; the direction angle is $13.42 ° .$

[T] Two forces, a vertical force of $26$

lb and another of $45$

lb, act on the same object. The angle between these forces is $55 ° .$

Find the magnitude and direction angle from the positive x-axis of the resultant force that acts on the object. (Round to two decimal places.)

[T] Three forces act on object. Two of the forces have the magnitudes $58$

N and $27$

N, and make angles $53 °$

and $152 °,$

respectively, with the positive x-axis. Find the magnitude and the direction angle from the positive x-axis of the third force such that the resultant force acting on the object is zero. (Round to two decimal places.)

The magnitude of the third vector is $60.03$

N; the direction angle is $259.38 ° .$

Three forces with magnitudes $80$

lb, $120$

lb, and $60$

lb act on an object at angles of $45 °,$

60 °

and $30 °,$

respectively, with the positive x-axis. Find the magnitude and direction angle from the positive x-axis of the resultant force. (Round to two decimal places.)

This figure is the first quadrant of a coordinate system. There are three vectors, all with the origin as the initial point. The first vector is labeled “60 lb” and makes an angle of 30 degrees with the x-axis. The second vector is labeled “80 lb” and makes an angle of 45 degrees with the x-axis. The third vector is labeled “120 lb” and makes a 60 degree angle with the x-axis.

[T] An airplane is flying in the direction of $43 °$

east of north (also abbreviated as $N 43 E)$

at a speed of $550$

mph. A wind with speed $25$

mph comes from the southwest at a bearing of $N 15 E .$

What are the ground speed and new direction of the airplane?

This figure is the first quadrant of a coordinate system. There are two vectors both of which have the origin as the initial point. The first vector is labeled “550 miles per hour” and has an angle of 43 degrees from the y-axis. There is also an image of an airplane at the end of the vector. The second vector is labeled “25 miles per hour” and has an angle of 15 degrees from the y-axis.

The new ground speed of the airplane is $572.19$

mph; the new direction is $N 41.82 E .$

[T] A boat is traveling in the water at $30$

mph in a direction of $N 20 E$

(that is, $20 °$

east of north). A strong current is moving at $15$

mph in a direction of $N 45 E .$

What are the new speed and direction of the boat?

This figure is the first quadrant of a coordinate system. There are two vectors both of which have the origin as the initial point. The first vector is labeled “15” and has an angle of 45 degrees from the y-axis. The second vector is labeled “30” and has an angle of 20 degrees from the y-axis. There is also an image of a boat at the end of the vector.

[T] A 50-lb weight is hung by a cable so that the two portions of the cable make angles of $40 °$

and $53 °,$

respectively, with the horizontal. Find the magnitudes of the forces of tension $T_{1}$

and $T_{2}$

in the cables if the resultant force acting on the object is zero. (Round to two decimal places.)

This figure is a horizontal line with two vectors below the line formina a triangle with the line. The vectors point toward the ends of the line. The angle between the horizontal line and the first vector is 40 degrees. The angle between the horizontal line and the second vector is 53 degrees. At the point where the two vectors meet is a rectangle labeled “50 lb.”

‖ T_{1} ‖ = 30.13 lb,

‖ T_{2} ‖ = 38.35 lb

[T] A 62-lb weight hangs from a rope that makes the angles of $29 °$

and $61 °,$

respectively, with the horizontal. Find the magnitudes of the forces of tension $T_{1}$

and $T_{2}$

in the cables if the resultant force acting on the object is zero. (Round to two decimal places.)

[T] A 1500-lb boat is parked on a ramp that makes an angle of $30 °$

with the horizontal. The boat’s weight vector points downward and is a sum of two vectors: a horizontal vector $v_{1}$

that is parallel to the ramp and a vertical vector $v_{2}$

that is perpendicular to the inclined surface. The magnitudes of vectors $v_{1}$

and $v_{2}$

are the horizontal and vertical component, respectively, of the boat’s weight vector. Find the magnitudes of $v_{1}$

and $v_{2} .$

(Round to the nearest integer.)

This figure shows a right triangle. The angle between the horizontal base and the hypotenuse is 30 degrees. On the hypotenuse is the image of a boat. From center of the boat there are three vectors. Two of the vectors are orthogonal with one in the direction of the boat and the other below the boat. The third vector is down, perpendicular to the vertical side.

‖ v_{1} ‖ = 750

lb, $‖ v_{2} ‖ = 1299$

[T] An 85-lb box is at rest on a $26 °$

incline. Determine the magnitude of the force parallel to the incline necessary to keep the box from sliding. (Round to the nearest integer.)

A guy-wire supports a pole that is $75$

ft high. One end of the wire is attached to the top of the pole and the other end is anchored to the ground $50$

ft from the base of the pole. Determine the horizontal and vertical components of the force of tension in the wire if its magnitude is $50$

lb. (Round to the nearest integer.)

This figure is a tower with a guy wire from the top to the ground. The tower, guy wire, and the ground form a right triangle. The base is labeled “50 feet” and is horizontal. The other side is labeled “75 feet” and is vertical. This side is the tower.

The two horizontal and vertical components of the force of tension are $28$

lb and $42$

lb, respectively.

A telephone pole guy-wire has an angle of elevation of $35 °$

with respect to the ground. The force of tension in the guy-wire is $120$

lb. Find the horizontal and vertical components of the force of tension. (Round to the nearest integer.)

Vectors in the Plane

Vector Representation

Combining Vectors

Vector Components

Proof of Commutative Property

Proof of Distributive Property

Unit Vectors

Applications of Vectors

Key Concepts

Glossary