Vectors in Two Dimensions

Introduction

In engineering, physics, and mathematics, vectors are a mathematical or graphical representation of a physical quantity that has a magnitude as well as a direction. Both magnitude and direction are required to define a vector. A force vector, for example, will have both a magnitude (a scalar quantity such as 10 Newtons) and a direction (up, down, left, right, 30o from the horizontal, etc.). While using vectors in three dimensional space is more applicable to the real world, it is far easier to learn vectors in two dimensional space first. This handout will only focus on vectors in two dimensions.

In two dimensional space, (R2), a vector can be represented graphically as an arrow with a starting point and an ending point. The length of the arrow represents the magnitude of the vector, while the direction in which the arrow is pointing represents the vector’s direction. The negative, or opposite of a vector is a vector with the same magnitude as the original, and the opposite direction. In this handout, vectors will be distinguished by using a bold font, for example, the force vector: . The

magnitude of a vector will be written as the absolute value of that vector: | |. The figure below𝑭𝑭 2 shows an arbitrary vector in R , where its magnitude is defined by its length𝑭𝑭 , and its direction is defined by the angle ฀. 𝑭𝑭

𝑦𝑦

𝑭𝑭

฀

𝑥𝑥

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Adding Vectors Geometrically

Vectors can be added using one of two methods: the parallelogram method or the tail-to-tip method. Vector addition is commutative; therefore, + = + .

𝑨𝑨 𝑩𝑩 𝑩𝑩 𝑨𝑨 To add vectors using the parallelogram method, two arbitrary vectors (A and B) can be moved so that their tails are coincident (their tales share the same point). A parallelogram is then drawn using the two vectors. The sum of the two vectors (R) is the vector drawn from the coincident tails to the opposite vertex of the parallelogram.

Since a vector is defined by its magnitude and direction, vectors can be translated to any position as long as the magnitude and direction of the vector remain the same.

Subtracting a vector from another can be seen as adding a negative vector.

Example A: + =

𝑨𝑨 𝑩𝑩 𝑹𝑹

𝑹𝑹 𝑩𝑩

𝑩𝑩 𝑩𝑩 𝑨𝑨 𝑨𝑨 𝑨𝑨

Example B: + ( ) =

𝑨𝑨 −𝑩𝑩 � 𝑜𝑜𝑜𝑜 𝑨𝑨 −𝑩𝑩 � 𝑹𝑹

𝑩𝑩 𝑩𝑩

𝑩𝑩 𝑨𝑨 𝑨𝑨 𝑨𝑨 −𝑩𝑩 𝑹𝑹 Provided by the Academic Center for Excellence 2 Vectors in Two Dimensions January 2017

To add two vectors using the tail-to-tip method, take the tail of one vector ( ), and move it so it is

coincident with the tip of the other vector ( ). Then draw a vector with its tail𝑩𝑩 coincident with the tail of the first vector ( ) and its tip coincident𝑨𝑨 with the tip of the second vector ( ). This vector is the resultant vector ( 𝑨𝑨). 𝑩𝑩 𝑹𝑹 Example C: + =

𝑨𝑨 𝑩𝑩 𝑹𝑹

𝑩𝑩 𝑩𝑩 𝑹𝑹 𝑩𝑩

𝑨𝑨 𝑨𝑨 𝑨𝑨

When trying to add three or more vectors together, it is possible to use the parallelogram method. First add two vectors, then add the resultant to the third vector. However, this method can become somewhat hectic and time consuming; therefore, the tail-to-tip method is preferred.

To add three or more vectors using the tail-to-tip method, follow the steps in the previous problem while adding another vector.

Example D: + + =

𝑩𝑩 𝑨𝑨 𝑪𝑪 𝑹𝑹 A A

B C C C B B

A R

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Components of Vectors

A useful way to describe the direction or orientation of a vector is by projecting it onto a set of axes. These axes can be arbitrary, but it is often easier to use the principal axes (the , and -axis). The

process of projecting a vector onto the principal axes is often called resolving the𝑥𝑥 vector𝑦𝑦 into𝑧𝑧 its components. This process can be thought of as finding the vector’s shadow on the principal axis.

Example E: Find the components of the given vector, F

𝑦𝑦

Step One 𝑭𝑭 Draw a two dimensional vector defined by its magnitude , and its direction, ฀. The answer to this example will change depending on which quadrant the vector is𝑭𝑭 drawn in. In this example the vector is drawn in the first quadrant and the direction is measured counterclockwise from the -axis. ฀

𝑥𝑥 𝑥𝑥

𝑦𝑦 Step Two 𝑭𝑭 To resolve this vector into its components projected onto the - and -axis, draw its “shadow” on the - and -axis ( and , respectively). 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝒙𝒙 𝒚𝒚 𝑭𝑭𝒚𝒚 Using𝑦𝑦 the parallelogram𝑭𝑭 𝑭𝑭 law of vector addition, it ฀ can be seen that and add to equal .

𝒙𝒙 𝒚𝒚 𝑭𝑭 𝑭𝑭 𝑭𝑭

𝑥𝑥 𝑭𝑭𝒙𝒙

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Example E: (continued):

𝑦𝑦 Step Three Since vectors can be moved from one position to another as long as the magnitude and direction are not altered, move either component of to form a right triangle with the 𝑭𝑭 original vector . 𝑭𝑭 In this example𝑭𝑭 the -component was moved to form a right ฀ 𝑭𝑭𝒚𝒚 triangle with . Moving the - component would also work, but it would require 𝑦𝑦more calculations and produce the answer 𝑭𝑭 𝑥𝑥 in a more complex form. 𝑥𝑥 𝑭𝑭𝒙𝒙

Step Four Since the length of a vector is represented by its magnitude, right triangle trigonometry can be used to find the magnitude

of and in terms of and ฀, right triangle trigonometry is used: 𝑭𝑭 𝑭𝑭𝒙𝒙 𝑭𝑭𝒚𝒚 𝑭𝑭 𝒚𝒚 𝑭𝑭 = (฀) = = Definition of cosine

��∗ 𝑭𝑭 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑭𝑭𝒙𝒙

𝑐𝑐𝑜𝑜𝑐𝑐 ℎ𝑦𝑦𝑦𝑦𝑦𝑦𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑦𝑦𝑦𝑦𝑎𝑎 𝑭𝑭 𝑐𝑐𝑠𝑠𝑠𝑠 Multiply both sides of

( ( ฀) =

฀ ฀ the equation by F and

) 𝒙𝒙 𝑭𝑭 then simplify 𝑭𝑭 ∗ 𝑐𝑐𝑜𝑜𝑐𝑐 ∗ 𝑭𝑭 𝑭𝑭

= (฀) = (฀) Solution

𝒙𝒙 𝑭𝑭 𝑭𝑭 ∗ 𝑐𝑐𝑜𝑜𝑐𝑐 𝑭𝑭𝒙𝒙 𝑭𝑭 ∗ 𝑐𝑐𝑜𝑜𝑐𝑐

The same method is used to find that = (฀) for this specific process and vector. 𝒚𝒚 𝑭𝑭 𝑭𝑭 ∗ 𝑐𝑐𝑠𝑠𝑠𝑠

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Example F: Write the magnitude of the vector used in Example E, in terms of its

components: and . 𝑭𝑭 𝑭𝑭𝒙𝒙 𝑭𝑭𝒚𝒚

𝑦𝑦

Step One Recall that the and components of a vector can 𝑭𝑭 be moved to produce a right triangle with the

original vector.𝑥𝑥 𝑦𝑦 ฀ 𝑭𝑭𝒚𝒚

𝑥𝑥 𝑭𝑭𝒙𝒙

Step Two Because the magnitude of a vector is represented by = ( ) + ( ) its length, the Pythagorean Theorem can be used to 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝒙𝒙 𝒚𝒚 calculate the magnitude of vector . 𝑭𝑭 𝑭𝑭 𝑭𝑭 𝑭𝑭

Solution Solve for , then simplify to find the solution. Since | | = ( ) + ( ) the magnitude of is written as | |, the solution is 𝟐𝟐 𝟐𝟐 𝒙𝒙 𝒚𝒚 𝑭𝑭 | | 𝑭𝑭 � 𝑭𝑭 𝑭𝑭 written in the form “ =…”. 𝑭𝑭 𝑭𝑭 𝑭𝑭

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Example G: Write the direction of the vector used in Example E and F, in terms of its components and . 𝑭𝑭 𝑭𝑭𝒙𝒙 𝑭𝑭𝒚𝒚

𝑦𝑦

Step One The and components of are moved to form a right triangle. 𝑭𝑭 𝑥𝑥 𝑦𝑦 𝑭𝑭

฀ 𝑭𝑭𝒚𝒚

𝑥𝑥 𝑭𝑭𝒙𝒙

Step Two tan(฀) = Recall the definition of tangent in a right triangle.

𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑐𝑐𝑠𝑠𝑜𝑜𝑜𝑜 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑐𝑐𝑜𝑜𝑠𝑠𝑜𝑜

Step Three tan(฀) = Substitute the given values into the equation of tangent. 𝒚𝒚 𝑭𝑭

𝑭𝑭𝑥𝑥

Step Four (tan(฀)) = tan ( ) Take the inverse tangent of both −1 −1 𝑭𝑭𝒚𝒚 sides of the equation, then simplify. 𝑜𝑜𝑎𝑎𝑠𝑠 𝑭𝑭𝑥𝑥

฀ = tan ( ) Solution −1 𝑭𝑭𝒚𝒚 𝑭𝑭𝑥𝑥 Provided by the Academic Center for Excellence 7 Vectors in Two Dimensions January 2017

Unit Vectors

A unit vector is a vector with a magnitude of one. Unit vectors are used to describe a direction, but not a specific magnitude. In this handout, unit vectors will be written the same way as regular vectors, but with a hat on top of it them. The hat signifies that the vector has a magnitude of one. An example of writing a unit vector with a hat on top of it would be the unit vector .

𝒖𝒖� Since describing the direction of the - and -axes is useful, there are special unit vectors that

describe these directions. The unit vector𝑥𝑥 that𝑦𝑦 describes the -direction is the vector, and the unit vector that describes the -direction is the vector. 𝑥𝑥 𝒊𝒊̂ 𝑦𝑦 𝒋𝒋̂ To calculate a vector that points in the direction of an arbitrary vector but has a magnitude of one, divide each individual component by the magnitude of the vector.

The unit vector of an arbitrary vector can be written mathematically as = | | 𝑭𝑭 𝑭𝑭 𝒖𝒖� 𝑭𝑭 Example H: Find the unit vector that points in the same direction as the velocity vector = 3 + 4

𝑽𝑽 𝒊𝒊̂ 𝒋𝒋 ̂ Step One | | = 3 + 4 = 5 Use the Pythagorean Theorem to

2 2 find that the magnitude of is 5. 𝑽𝑽 � 𝑽𝑽 3 + 4 Step Two

= Divide the vector V by its 5 𝒊𝒊̂ 𝒋𝒋̂ magnitude, 5, then simplify. 𝒖𝒖�

3 4 = + 5 5 Solution

𝒖𝒖� 𝒊𝒊̂ 𝒋𝒋̂

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Cartesian Vector Notation

The unit vectors and are very useful and can be used to write a vector in terms of its magnitude

and direction. For𝒊𝒊̂ example,𝒋𝒋̂ say there is a force vector, , described by its components = 25 and = 10 . 𝑭𝑭 𝑭𝑭𝒙𝒙 𝑁𝑁 𝑭𝑭𝒚𝒚 𝑁𝑁 Utilizing vector addition, and the fact that + = , the vector can be rewritten in a way that uses unit vectors. The first step in this process𝑭𝑭𝒙𝒙 is𝑭𝑭 to𝒚𝒚 take𝑭𝑭 each of the components, and multiply their given magnitude by the unit vector that describes their directions; for this example, multiply the magnitude of the -component of by the unit vector that describes the -direction to get 25

(or 25 ) for the -component𝑥𝑥 . The𝑭𝑭 same is done to rewrite the -component𝑥𝑥 as 10 . ∗ 𝒊𝒊̂ 𝒊𝒊̂ 𝑥𝑥 𝑦𝑦 𝒋𝒋̂ = (25 ) = (10 )

𝑭𝑭𝒙𝒙 𝒊𝒊̂ 𝑁𝑁 𝑭𝑭𝒚𝒚 𝒋𝒋̂ 𝑁𝑁 Since the components of add to equal , the vector can be written in terms of its components:

𝑭𝑭 𝑭𝑭 = (25 + 10 ) .

̂ ̂ This form of writing a vector is called Cartesian𝑭𝑭 Vector𝒊𝒊 Notation.𝒋𝒋 𝑁𝑁

An alternate way to write a vector in terms of its components is a by using angle brackets to contain the values of the components. The alternate notation does not use Cartesian Vectors to denote which value corresponds to which component, instead it uses a strict format. The format for the alternate notation is: = < , >.

𝑭𝑭 𝑭𝑭𝒙𝒙 𝑭𝑭𝒚𝒚 For example, the vector = (25 + 10 ) can be written as = < 25, 10 > using the alternate notation. 𝑭𝑭 𝒊𝒊̂ 𝒋𝒋̂ 𝑁𝑁 𝑭𝑭 𝑁𝑁

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Example I: Given a force, , with a magnitude of 62 pounds and a direction 50 degrees clockwise from the positive -axis,𝑭𝑭 write the components of in Cartesian Vector Notation. 𝑥𝑥 𝑭𝑭

𝑦𝑦 Step One 50o Begin by drawing the vector in the 𝑥𝑥 -plane 𝑥𝑥𝑦𝑦 62lb

y Step Two Find the magnitude and direction of each = 39.9 component of : 𝒙𝒙 x

𝑭𝑭 𝑁𝑁 o 𝑁𝑁 50 : Magnitude𝑭𝑭 : 62 (50 ) = 39.9 .5 𝑦𝑦

47 Direction: positive -direction, or the 𝒙𝒙 𝑭𝑭 direction∗ 𝑐𝑐𝑜𝑜𝑐𝑐 𝑁𝑁 𝑁𝑁

=

𝒙𝒙 𝑥𝑥 𝒊𝒊̂ 𝑭𝑭 62lb : Magnitude: 62 (50 ) = 47.5 Direction: negative -direction,𝑦𝑦 or the – 𝒚𝒚 𝑭𝑭 direction∗ 𝑐𝑐𝑠𝑠𝑠𝑠 𝑁𝑁 𝑁𝑁 𝑦𝑦 𝒋𝒋̂

Solution By finding the magnitude and direction of each = 39.9 47.5 individual component, this vector can be written in Cartesian Vector Notation. 𝑭𝑭 𝒊𝒊̂ − 𝒋𝒋̂

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Adding Vectors Mathematically

While adding vectors using the parallelogram and tail-to-tip method is useful because it provides a general idea of what the result of adding two or more vectors will look like, vectors can also be added mathematically by utilizing Cartesian Vector Notation.

Example J: Given two vectors written in Cartesian Vector Notation: = 2 + 4 and = 5 +

3 , find the sum of and . 𝑭𝑭 𝒊𝒊̂ 𝒋𝒋̂ 𝑷𝑷 𝒊𝒊̂ 𝒋𝒋̂ 𝑭𝑭 𝑷𝑷

Step One Rewrite the vectors in terms of their + = ( + ) + + = components. The sum of and will be the 𝑭𝑭 𝑷𝑷 𝑭𝑭𝒙𝒙 𝑭𝑭𝒚𝒚 �𝑷𝑷𝒙𝒙 𝑷𝑷𝒚𝒚� 𝑹𝑹

resultant vector . 𝑭𝑭 𝑷𝑷 𝑹𝑹

Step Two + = (2 + 4 ) + (5 + 3 ) = Substitute in the values for and , written in

𝑭𝑭 𝑷𝑷 𝒊𝒊̂ 𝒋𝒋̂ 𝒊𝒊̂ 𝒋𝒋̂ 𝑹𝑹 Cartesian Vector Notation, 𝑹𝑹then simplify𝑭𝑭 by adding the components of each vector; that is the -component of with the -component

of 𝑥𝑥, and the -component𝑭𝑭 of 𝑥𝑥 with the - component𝑷𝑷 of𝑦𝑦 . 𝑭𝑭 𝑦𝑦 𝑷𝑷 = 7 + 7 Solution

𝑹𝑹 𝒊𝒊̂ 𝒋𝒋̂

One very common mistake when adding vectors mathematically is to add the values of the - and - components together. This is something that should not be done because it defeats the

𝒙𝒙purpose𝒚𝒚 of breaking the vector into its components before adding them. Adding the - and - components is much like adding 2 and 3 , the result is 2 + 3 , not 5 or 5 . 𝑥𝑥 𝑦𝑦

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Scalar Multiplication

One quality of vectors is that they can be multiplied by a scalar. Multiplying a vector by a scalar results in another vector that has an increased magnitude and/or a negated direction.

Example K: Given the vector = 5 + 6 , increase its magnitude by a factor of 5.

𝒗𝒗 𝒊𝒊̂ 𝒋𝒋̂

Step One 5 = 5(5 + 6 ) Multiply the vector by the scalar value 5, then

𝒗𝒗 𝒊𝒊̂ 𝒋𝒋̂ simplify by using the distributive property of multiplication

.5 = (25 + 30 ) Solution

𝒗𝒗 𝒊𝒊̂ 𝒋𝒋̂ Example L: Reverse the direction of 5 without changing its magnitude.

𝒗𝒗 Step One 5 = 1(25 + 30 ) Multiply the vector by the scalar value -1,

− 𝒗𝒗 − 𝒊𝒊̂ 𝒋𝒋̂ then simplify.

5 = ( 25 30 ) Solution

− 𝒗𝒗 − 𝒊𝒊̂ − 𝒋𝒋̂

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The Dot Product

The operations of vector addition and scalar multiplication result in vectors. By contrast, multiplying two vectors together using the dot product results in a scalar. The dot product is written mathematically as (reads “ dot ”), where and are both vectors.

𝑼𝑼 ∙ 𝑽𝑽 𝑼𝑼 𝑽𝑽 𝑼𝑼 𝑽𝑽 The dot product can only be used to multiply two vectors; it cannot be used to multiply a scalar and a vector, or a scalar and a scalar.

The dot product has many uses, such as finding the angle between two vectors, the projection of one vector onto another, the amount of work a force does on moving an object, etc. This handout will only go over how to calculate the dot product, not how to apply the dot product do various problems.

There are 2 main formulas used to calculate the dot product of two vectors:

Equation 1: Where and are arbitrary vectors, and = + , and = + ,

𝑼𝑼 𝑽𝑽 𝑼𝑼 𝑼𝑼𝒙𝒙 𝑼𝑼𝒚𝒚 𝑽𝑽 𝑽𝑽𝒙𝒙 𝑽𝑽𝒚𝒚 = +

𝑼𝑼 ∙ 𝑽𝑽 𝑼𝑼𝒙𝒙 ∗ 𝑽𝑽𝒙𝒙 𝑼𝑼𝒚𝒚 ∗ 𝑽𝑽𝒚𝒚 Equation 2: Where ฀ is the angle between and ,

𝑼𝑼 𝑽𝑽 = | | | | (฀)

𝑼𝑼 ∙ 𝑽𝑽 𝑼𝑼 ∗ 𝑽𝑽 ∗ 𝑐𝑐𝑜𝑜𝑐𝑐

To determine which equation to use, review the given information, then determine which equation is the simplest to use. If the components of two vectors are given, the first equation is preferred in calculating the dot product. Likewise, if the magnitude and direction of the two vectors are given, the second equation is preferred.

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Example M: Given that = 10 + 2 , and = 4 + 3 , find .

𝑼𝑼 𝒊𝒊̂ 𝒋𝒋̂ 𝑽𝑽 𝒊𝒊̂ 𝒋𝒋̂ 𝑼𝑼 ∙ 𝑽𝑽

Step One

Because the magnitude of the components of vectors and is = + the only information given, equation 1 used. 𝑼𝑼 𝑽𝑽 𝑼𝑼 ∙ 𝑽𝑽 𝑼𝑼𝒙𝒙 ∗ 𝑽𝑽𝒙𝒙 𝑼𝑼𝒚𝒚 ∗ 𝑽𝑽𝒚𝒚

Step Two = (10)(4) + (2)(3) Substitute the given information into the equation, then simplify.

𝑼𝑼 ∙ 𝑽𝑽

= 46 Solution

𝑼𝑼 ∙ 𝑽𝑽

Example N: Given that has a magnitude of 20N and a direction 20o to the right of the positive

o -axis, and𝑭𝑭 that has a magnitude of 10N with a direction of 32 above the positive 𝑦𝑦-axis, find 𝑷𝑷 . 𝑥𝑥 𝑭𝑭 ∙ 𝑷𝑷

𝑦𝑦 Step One 𝑭𝑭 Begin by drawing the two vectors in 20o the x-y plane

32o 𝑷𝑷

𝑥𝑥

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Example N(continued):

𝑦𝑦 Step Two 𝑭𝑭 Find that the angle between vectors F and P is equal to 90 (32 + 20 ) = 38 38o 𝑦𝑦 𝑦𝑦 𝑦𝑦 𝑦𝑦 −

𝑷𝑷

𝑥𝑥 Step Three Since the magnitude of both vectors, = | | | | (฀) and the angle between the two vectors are known, use the second equation. 𝑭𝑭 ∙ 𝑷𝑷 𝑭𝑭 ∗ 𝑷𝑷 ∗ 𝑐𝑐𝑜𝑜𝑐𝑐

Step Four = (20 )(10 ) (38 ) Substitute information into 𝑦𝑦 equation, then simplify. 𝑭𝑭 ∙ 𝑷𝑷 𝑁𝑁 𝑁𝑁 ∗ 𝑐𝑐𝑜𝑜𝑐𝑐

= 157.6 Solution

𝑭𝑭 ∙ 𝑷𝑷

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