GEOMETRY IN SPACE AND VECTORS: THE CROSS PRODUCT

MAKALAHUNTUK MEMENUHI TUGAS KULIAHAdvanced CalculusYang dibina oleh Ibu Trianingsih Eni Lestari, M.Sc

olehAgus Ahmad Rizqi 140311601706 Arief Nugroho 140311602668

UNIVERSITAS NEGERI MALANGFAKULTAS MATEMATIKA DAN ILMU PENGETAHUAN ALAMJURUSAN MATEMATIKANovember 2015

PREFACEThanks to Almighty God who has given His bless to the writers for finishing the paper about the translation and rotation of axes for completing assignment of Advanced Calculus course that lectured by Mrs. Trianingsih Eni Lestari with no meaningful obstacles. The writers also wish to express his deep and sincere gratitude for those who have guided us in completing this paper on time. This paper contains some discussion about continuing material of geometruy in space and vectors, the cross product and also supply some exercises and their solutions that writers compile from some reference and from our discussion about these materials. Hopefully, this paper can be useful for the readers that want to learn deeply about the materials served here.

1. BackgroundQuantities such as velocity, force, torque, and displacement, rquire both a magnitude and a direction for complete spesification. Such quantities is called vectors and represented by arrows. The length of the arrow reprresents the magnitude, or length, of the vector, its direction is the direction of the vector. A vector can be multiplied by scalar, or added and subtracted by other vectors. Multiplication of two vectors u and v, symbolized by u.v, is called the dot product or scalar product, resulting a scalar. The dot product can be applied for finding angle between two vectors and finding projection of a vector to another one. Beside the dot product, there is another operation on vectors called by cross product symbolized by u v.2. PurposeOur paper aims to complete the task of subject Advanced Calculus given by mrs. Trianingsih Eni Lestari. Moreover, the paper also made to help us and all of the readers to know more about the cross product. This paper also aims to help the readers to solve the problem or some exercises in the book with see our examples of exercises that we have done, of course, the cross product.3. DiscussionCross product of 2 vectors u and v where u = { u1 , u2 , u3 } and v = { v1 , v2 , v3 } defined byu x v = { u2 v3 - v2 u3, u3 v1 - u1 v3 , u1 v2 - u2 v1}This formula form is hard to remember, so we can use determinants to help us finding cross product. Remember that cross product of two vectors is vector.The determinants value of 2x2 matrix is

The determinants value of 3x3 matrix is

Using determinants we define u x v as

Note the right vector u placed in the second row, and the left vector v placed in the third row. If we change that position, it will change the value of determinants and cross product become:

It called anticommutative law.Example 1:Let u = { 1, -2 , -1 } and v = { -2, 4, 1 } calculate and using determinants definitionsAnswer :

Geometric interpretation of Theorem A : Let are vectors in three space , and is angle between them, then:1. u.(u x v) = 0 = v.(u x v) , that is u x v is perpendicular to both u and v .2. u, v,and u x v form a right- handed triple.3. =

Proof : 1. u.(u x v) = u1 (u2 v3 - v2 u3)+ u2 (u3 v1 - u1 v3 ) + u3 (u1 v2 - u2 v1) = 02. meaning of right- handed triple showed in figure 1.Where between u and v . its like u that rotate trough v, so u coincide with v. 3. We need langranges identity = = = = = It is important that we have geometric interpretations of both u.v and u v. While both products were originally defined in terms of components that depend on a choice of coordinate system, they are actually independent of coordinate systems. They are intrinsic geometric quantities, and it will yields the same results for u.v and u v no matter how the coordinates used to compute them introduced.Theorem BTwo vectors u and v in three space are parallel if and only if u v = 0We note that two vectors will be parallel if and only if the angle between them is either 0 or , that is sin = 0. So

There some applications of croos product. The first, is to find the equation of the plane through three noncollinear points. Example 2. Find the equation of the plane (Figure 2) through three noncollinear points P1(1, -2,3), P2(4,1,-2), and P3(-2,-3,0).Solution. Let u and vu v= is perpendicular to both u and v and thus to the plane containing them. The plane through (4,1,-2) with normal 14i 24j- 6k has equation

Or

Eample 3. Show that the area of a parallelogram with a and b as adjacent sides is a b.SolutionThe area of a parallelogram is the product of the base times the height.The base is the magnitude of a, that is a, and the height is equal to bsin. So the area, A=base height

Example 4Show that the volume of the parallelepiped determined by the vectors a, b, and c is

Solution.

Refer to figure 4 and regard the parallelogram determined by b and c as the base of the parallelepiped. The area of this base is b x c and the height h of the paralellepiped is the absolute value of the scalar projection of a on b c. Thus,

and There are algebraic properties consisting rules for calculating with cross products.Theorem CIf u, v, and w are vectors in three space and k is a scalar, then1. u v = -(v u) (anticommutative law)2. u (v + w) = (u v) + (u w) (left distributive law)3. k (u v) = (ku) v = u (kv)4. u 0 = 0 u = 0, u u = 05. (u v).w = u.(v w)6. u (v w) = (u.w)v (u.v)wBased on the theorem, we get a simple way to calculate the cross product by using fact that i j = k, j k =i and k i = j.Example 5Calculate u v if u = 3i 2j + k and v = 4i + 2j -3k.Solution.

4. Solution of some exercises Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

5. ConclusionCross product of 2 vectors u and v where u = { u1 , u2 , u3 } and v = { v1 , v2 , v3 } defined by u x v = { u2 v3 - v2 u3, u3 v1 - u1 v3 , u1 v2 - u2 v1}.If are vectors in three space , and is angle between them, then:1. u.(u x v) = 0 = v.(u x v) , that is u x v is perpendicular to both u and v .2. u, v,and u x v form a right- handed triple. = Moreover, two vectors u and v in three space are parallel if and only if u v = 0Some application of cross product is to find equation of a plane through three noncollinear points, to find area of parallelogram aand triangle, and to find volume of parallelepiped. For calculating the cross product we also can use some algebraic properties, that is, if u, v, and w are vectors in three space and k is a scalar, then1. u v = -(v u) (anticommutative law)2. u (v + w) = (u v) + (u w) (left distributive law)3. k (u v) = (ku) v = u (kv)4. u 0 = 0 u = 0, u u = 05. (u v).w = u.(v w)6. u (v w) = (u.w)v (u.v)wBased on the theorem, we get a simple way to calculate the cross product by using fact that i j = k, j k =i and k i = j.

REFERENCEPurcelle & Verberg, Dale. 2002. Calculus Ninth Editions.