Vectors in Two Dimensions

Transcription

1 Vectors in Two Dimensions Introduction In engineering, phsics, and mathematics, vectors are a mathematical or graphical representation of a phsical quantit that has a magnitude as well as a direction. oth magnitude and direction are required to define a vector. force vector, for eample, will have both a magnitude (a scalar quantit such as 10 Newtons) and a direction (up, down, left, right, 30 o 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 onl focus on vectors in two dimensions. In two dimensional space, (R 2 ), a vector can be represented graphicall 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 b using a bold font, for eample, the force vector:. The magnitude of a vector will be written as the absolute value of that vector:. The figure below shows an arbitrar vector in R 2, where its magnitude is defined b its length, and its direction is defined b the angle. Provided b the cademic Center for Ecellence 1 Vectors in Two Dimensions

2 dding Vectors Geometricall Vectors can be added using one of two methods: the parallelogram method or the tail-to-tip method. Vector addition is commutative; therefore, + = +. To add vectors using the parallelogram method, two arbitrar vectors ( and ) can be moved so that their tails are coincident (their tales share the same point). 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 verte of the parallelogram. Since a vector is defined b its magnitude and direction, vectors can be translated to an 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. Eample : + = R R Eample : oo + ( ) = R R Provided b the cademic Center for Ecellence 2 Vectors in Two Dimensions

3 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 tail coincident with the tail of the first vector () and its tip coincident with the tip of the second vector (). This vector is the resultant vector (R). Eample C: + = R R When tring to add three or more vectors together, it is possible to use the parallelogram method. irst 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. Eample D: + + C = R C C C R Provided b the cademic Center for Ecellence 3 Vectors in Two Dimensions

4 Components of Vectors useful wa to describe the direction or orientation of a vector is b projecting it onto a set of aes. These aes can be arbitrar, but it is often easier to use the principal aes (the, and z-ais). The process of projecting a vector onto the principal aes is often called resolving the vector into its components. This process can be thought of as finding the vector s shadow on the principal ais. Eample E: ind the components of the given vector, Draw a two dimensional vector defined b its magnitude, and its direction,. The answer to this eample will change depending on which quadrant the vector is drawn in. In this eample the vector is drawn in the first quadrant and the direction is measured counterclockwise from the -ais. To resolve this vector into its components projected onto the - and -ais, draw its shadow on the and -ais ( and, respectivel). Using the parallelogram law of vector addition, it can be seen that and add to equal. Provided b the cademic Center for Ecellence 4 Vectors in Two Dimensions

5 Eample E: (continued): 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 original vector. In this eample the -component was moved to form a right triangle with. Moving the - component would also work, but it would require more calculations and produce the answer in a more comple form. Step our Since the length of a vector is represented b its magnitude, right triangle trigonometr can be used to find the magnitude of and in terms of and, right triangle trigonometr is used: = css() coc() = aaaaaaaa haaaaa = coc() = Definition of cosine Multipl both sides of the equation b and then simplif = coc() = coc() The same method is used to find that = css() for this specific process and vector. Provided b the cademic Center for Ecellence 5 Vectors in Two Dimensions

6 Eample : Write the magnitude of the vector used in Eample E, in terms of its components: and. Recall that the and components of a vector can be moved to produce a right triangle with the original vector. 2 = ( ) 2 + ( ) 2 ecause the magnitude of a vector is represented b its length, the Pthagorean Theorem can be used to calculate the magnitude of vector. = ( ) 2 + ( ) 2 Solve for, then simplif to find the solution. Since the magnitude of is written as, the solution is written in the form =. Provided b the cademic Center for Ecellence 6 Vectors in Two Dimensions

7 Eample G: Write the direction of the vector used in Eample E and, in terms of its components and. The and components of are moved to form a right triangle. tan() = oooocsoo aaaacoso Recall the definition of tangent in a right triangle. tan() = Step Three Substitute the given values into the equation of tangent. oas 1 (tan()) = tan 1 ( ) Step our Take the inverse tangent of both sides of the equation, then simplif. = tan 1 ( ) Provided b the cademic Center for Ecellence 7 Vectors in Two Dimensions

8 Unit Vectors 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 wa as regular vectors, but with a hat on top of it them. The hat signifies that the vector has a magnitude of one. n eample of writing a unit vector with a hat on top of it would be the unit vector u. Since describing the direction of the - and -aes is useful, there are special unit vectors that describe these directions. The unit vector that describes the -direction is the i vector, and the unit vector that describes the -direction is the j vector. To calculate a vector that points in the direction of an arbitrar vector but has a magnitude of one, divide each individual component b the magnitude of the vector. The unit vector of an arbitrar vector can be written mathematicall as u = Eample H: ind the unit vector that points in the same direction as the velocit vector V = 3i + 4j V = = 5 Use the Pthagorean Theorem to find that the magnitude of V is 5. u = 3i + 4j 5 Divide the vector V b its magnitude, 5, then simplif. u = 3 5 i j Provided b the cademic Center for Ecellence 8 Vectors in Two Dimensions

9 Cartesian Vector Notation The unit vectors i and j are ver useful and can be used to write a vector in terms of its magnitude and direction. or eample, sa there is a force vector,, described b its components = 25N and = 10N. Utilizing vector addition, and the fact that + =, the vector can be rewritten in a wa that uses unit vectors. The first step in this process is to take each of the components, and multipl their given magnitude b the unit vector that describes their directions; for this eample, multipl the magnitude of the -component of b the unit vector that describes the -direction to get 25 i (or 25i ) for the -component. The same is done to rewrite the -component as 10j. = (25i )N = (10j )N Since the components of add to equal, the vector can be written in terms of its components: = (25i + 10j )N. This form of writing a vector is called Cartesian Vector Notation. n alternate wa to write a vector in terms of its components is a b 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: = <, >. or eample, the vector = (25i + 10j )N can be written as = < 25, 10 > N using the alternate notation. Provided b the cademic Center for Ecellence 9 Vectors in Two Dimensions

10 Eample I: Given a force,, with a magnitude of 62 pounds and a direction 50 degrees clockwise from the positive -ais, write the components of in Cartesian Vector Notation. 50 o egin b drawing the vector in the -plane 62lb = 39.9N ind the magnitude and direction of each component of : = 47.5N 50 o 62lb : : Magnitude: 62 coc(50 )N = 39.9N Direction: positive -direction, or the i direction Magnitude: 62 css(50 )N = 47.5N Direction: negative -direction, or the j direction = 39.9i 47.5j finding the magnitude and direction of each individual component, this vector can be written in Cartesian Vector Notation. Provided b the cademic Center for Ecellence 10 Vectors in Two Dimensions

11 dding Vectors Mathematicall 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 mathematicall b utilizing Cartesian Vector Notation. Eample J: Given two vectors written in Cartesian Vector Notation: = 2i + 4j and P = 5i + 3j, find the sum of and P. + P = ( + ) + P + P = R Rewrite the vectors in terms of their components. The sum of and P will be the resultant vector R. + P = (2i + 4j ) + (5i + 3j ) = R R = 7i + 7j Substitute in the values for R and, written in Cartesian Vector Notation, then simplif b adding the components of each vector; that is the -component of with the -component of P, and the -component of with the component of P. One ver common mistake when adding vectors mathematicall is to add the values of the - and - components together. This is something that should not be done because it defeats the purpose of breaking the vector into its components before adding them. dding the - and - components is much like adding 2 and 3, the result is 2 + 3, not 5 or 5. Provided b the cademic Center for Ecellence 11 Vectors in Two Dimensions

12 Scalar Multiplication One qualit of vectors is that the can be multiplied b a scalar. Multipling a vector b a scalar results in another vector that has an increased magnitude and/or a negated direction. Eample K: Given the vector v = 5i + 6j, increase its magnitude b a factor of 5. 5v = 5(5i + 6j ). 5v = (25i + 30j ) Multipl the vector b the scalar value 5, then simplif b using the distributive propert of multiplication Eample L: Reverse the direction of 5v without changing its magnitude. 5v = 1(25i + 30j ) Multipl the vector b the scalar value -1, then simplif. 5v = ( 25i 30j ) Provided b the cademic Center for Ecellence 12 Vectors in Two Dimensions

13 The Dot Product The operations of vector addition and scalar multiplication result in vectors. contrast, multipling two vectors together using the dot product results in a scalar. The dot product is written mathematicall as U V (reads U dot V ), where U and V are both vectors. The dot product can onl be used to multipl two vectors; it cannot be used to multipl a scalar and a vector, or a scalar and a scalar. The dot product has man 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 onl go over how to calculate the dot product, not how to appl the dot product do various problems. There are 2 main formulas used to calculate the dot product of two vectors: Equation 1: Where U and V are arbitrar vectors, and U = U + U, and V = V + V, U V = U V + U V Equation 2: Where is the angle between U and V, U V = U V coc() 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. Provided b the cademic Center for Ecellence 13 Vectors in Two Dimensions

14 Eample M: Given that U = 10i + 2j, and V = 4i + 3j, find U V. U V = U V + U V U V = (10)(4) + (2)(3) ecause the magnitude of the components of vectors U and V is the onl information given, equation 1 used. Substitute the given information into the equation, then simplif. U V = 46 Eample N: Given that has a magnitude of 20N and a direction 20 o to the right of the positive -ais, and that P has a magnitude of 10N with a direction of 32 o above the positive -ais, find P. 20 o egin b drawing the two vectors in the - plane P 32 o Provided b the cademic Center for Ecellence 14 Vectors in Two Dimensions

15 Eample N(continued): 38 o P ind that the angle between vectors and P is equal to 90 ( ) = 38 P = P coc() Step Three Since the magnitude of both vectors, and the angle between the two vectors are known, use the second equation. P = (20N)(10N) coc(38 ) Step our Substitute information into equation, then simplif. P = Provided b the cademic Center for Ecellence 15 Vectors in Two Dimensions