8/22/2005 section2_3vector_algebra_empty.doc 1/ Vector Algebra. You understand scalar math, but what about vector mathematics?

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1 8/22/2005 section2_3vector_lgebra_empty.doc 1/3 2.3 Vector lgebra Reading ssignment: pp You understand scalar math, but what about vector mathematics? Consider, for example:.. C. D.. rithmetic Operations of Vectors Q: : HO: rithmetic Operations of Vectors. rithmetic Operations of Vectors and Scalars Say b is a scalar and is a vector. Q: What then is + b or b??

2 8/22/2005 section2_3vector_lgebra_empty.doc 2/3 : C. Multiplicative Operations of Vectors and Scalars Q: So, does the multiplication of scalar b and vector (i.e., b or b ) have any meaning? : HO: Multiplicative Operations of Vectors and Scalars We can now examine a super-important concept: HO: The Unit Vector D. Multiplicative Operations of Vectors Q: Can we multiply two vectors? :

3 8/22/2005 section2_3vector_lgebra_empty.doc 3/3 HO: The Dot Product HO: The Cross Product HO: The Triple Product E. Vectors lgebra Now that we know the rules of vector operations, we can analyze, manipulate, and simplify vector operations! HO: Example: Vector lgebra HO: Scalar, Vector, or Neither? F. Orthogonal and Orthonormal Vector Sets We can now use vector algebra to write equations that specify some relationship between sets of vectors. HO: Orthogonal and Orthonormal Vector Sets

4 8/19/2005 rithmetic Operations of Vectors.doc 1/4 rithmetic Operations of Vectors Vector ddition Consider two vectors, denoted and. Q: Say we add these two vectors together; what is the result? : The addition of two vectors results in another vector, which we will denote as C. Therefore, we can say: + = C The magnitude and direction of C is determined by the headto-tail rule. C=+

5 8/19/2005 rithmetic Operations of Vectors.doc 2/4 This is not a provable result, rather the head to-tail rule is the definition of vector addition. This definition is used because it has many applications in physics. For example, if vectors and represent two forces acting an object, then vector C represents the resultant force when and are simultaneously applied. C=+ Some important properties of vector addition: 1. Vector addition is commutative -> + = + C=+ C=+

6 8/19/2005 rithmetic Operations of Vectors.doc 3/4 2. Vector addition is associative -> (X+Y) + Z = X + (Y+Z) X X Y Y (X+Y)+Z Z X+(Y+Z) Z From these two properties, we can conclude that the addition of several vectors can be executed in any order: X Z Y Y X Z Z+Y+X = X+Y+Z Vector Subtraction First, we define the negative of a vector to be a vector with equal magnitude but opposite direction. -

7 8/19/2005 rithmetic Operations of Vectors.doc 4/4 Note that + (-) = 0 - We can therefore consider the addition of a negative vector as a subtraction: = 0 More generally, we can write: + (-) = E.G., Q: Is - = -? : What do you think?

8 8/19/2005 Multiplicative Operations of Vectors and Scalars.doc 1/4 Multiplicative Operations of Vectors and Scalars Consider a scalar quantity a and a vector quantity. We express the multiplication of these two values as: a = C In other words, the product of a scalar and a vector is a vector! Q: OK, but what is vector C? What is the meaning of a? : The resulting vector C has a magnitude that is equal to a times the magnitude of. In other words: C = a However, the direction of vector C is exactly that of. Therefore multiplying a vector by a scalar changes the magnitude of the vector, but not its direction.

9 8/19/2005 Multiplicative Operations of Vectors and Scalars.doc 2/4 For example: Note that + = 2.0! More generally, we find that scalar-vector multiplication is distributive as: a + b = (a +b) E.G., nd also distributive as: a + a C = a ( +C) E.G., C +C C

10 8/19/2005 Multiplicative Operations of Vectors and Scalars.doc 3/4 2 2(+C) 2C 3. Scalar-Vector multiplication is also commutative: a = a 4. Multiplication of a vector by a negative scalar is interpreted as: -a = a (-) E.G., = 2.0(-) 5. Division of a vector by a scalar is the same as multiplying the vector by the inverse of the scalar: 1 = a a

11 8/19/2005 Multiplicative Operations of Vectors and Scalars.doc 4/4 Scalar-Vector multiplication is likewise used in many physical applications. For example, say you start in Lawrence and head west at 70 mph for exactly 3.3 hours. Note your velocity has both direction (west) and magnitude (70 mph) it s a vector! Lets denote it as V = 70 mph west. Likewise, your travel time is a scalar; lets denote it as t = 3.3 h. Now, lets multiply the two together (i.e., t V). The magnitude of the resulting vector is 70(3.3) = 231 miles. The direction of the resulting vector is of course unchanged: west. vector describing a distance and a direction a directed distance! We find that t V = R, where R identifies your location after 3.3 hours! You re in Hays, Kansas! R = t V = 231 miles west

12 8/19/2005 The Dot Product.doc 1/5 The Dot Product The dot product of two vectors, and, is denoted as i. The dot product of two vectors is defined as: i = cos θ where the angle θ is the angle formed between the vectors and. θ 0 θ π IMPORTNT NOTE: The dot product is an operation involving two vectors, but the result is a scalar!! E.G.,: i The dot product is also called the scalar product of two vectors. = c

13 8/19/2005 The Dot Product.doc 2/5 Q1: So, what would the dot product of three vectors be? I.E.,: C i i =??? : Note also that the dot product is commutative, i.e.,: iβ = i Some more fun facts about the dot product include: 1. The dot product of a vector with itself is equal to the magnitude of the vector squared. 2 ΑΑ i = Α Αcos 0 = Α Therefore: Α = ΑiΑ

14 8/19/2005 The Dot Product.doc 3/5 2. If Αi = 0 (and Α 0, 0), then it must be true that: cos θ = 0 θ = 90 Thus, if Αi = 0, the two vectors are orthogonal (perpendicular). θ = 90 Αi = 0 3. If Αi = Α, then it must be true that: cos θ = 1 θ = 0 Thus, vectors and must have the same direction. They are said to be collinear (parallel). Αi = Α cos 0 = Α

15 8/19/2005 The Dot Product.doc 4/5 4. If Αi = Α, then it must be true that: cos θ = 1 θ = 180 Thus, vectors and point in opposite directions; they are said to be anti-parallel. Αi = Α cos 180 = Α 5. The dot product is distributive with addition, such that: ( + C) = + C For example, we can write: ( + ) ( + C) = ( + ) + ( + ) C = ( + ) + C ( + ) = + + C + C 2 = + + C + C (distributive) (commutative) (distributive)

16 8/19/2005 The Dot Product.doc 5/5 One application of the dot product is the determination of work. Say an object moves a distance d, directly from point P a to point P b, by applying a constant force F. Q: How much work has been done? First, we can specify the direct path from point P a to point P b with a directed distance: d P b Rab P a The work done is simply the dot product of the applied force vector and the directed distance! F P b θ FR Rab P a W = F R : = F Rab cosθ = d F ab cosθ FR FR The value F cosθ FR is said to be the scalar component of force F in the direction of directed distance R ab

17 8/19/2005 The Unit Vector.doc 1/2 The Unit Vector Now that we understand multiplication and division of a vector by a scalar, we can discuss a very important concept: the unit vector. Lets begin with vector. Say we divide this vector by its magnitude (a scalar value). We create a new vector, which we will denote as â : â = Q: How is vector â related to vector? : Since we divided by a scalar value, the vector â has the same direction as vector. ut, the magnitude of â is: â = = 1 The vector â has a magnitude equal to one! We call such a vector a unit vector. Jim Stiles The University of Kansas Dept. of EECS

18 8/19/2005 The Unit Vector.doc 2/2 unit vector is essentially a description of direction only, as its magnitude is always unit valued (i.e., equal to one). Therefore: * is a scalar value that describes the magnitude of vector. * â is a vector that describes the direction of. Rearranging the formula on the previous page, we can express vector as: = a ˆ Thus, any vector can be written as the product of its magnitude and its unit vector! = a ˆ â Q: What, approximately, is the magnitude of vector shown above? : = Jim Stiles The University of Kansas Dept. of EECS

19 8/19/2005 The Cross Product.doc 1/4 The Cross Product The cross product of two vectors, and, is denoted as. The cross product of two vectors is defined as: = ˆ sinθ a n Just as with the dot product, the angle θ is the angle between the vectors and. The unit vector a ˆn is orthogonal to both and (i.e., ˆ a = 0 and ˆ a = 0). n n θ x 0 θ π IMPORTNT NOTE: The cross product is an operation involving two vectors, and the result is also a vector. E.G.,: = C

20 8/19/2005 The Cross Product.doc 2/4 The magnitude of vector x is therefore: = sinθ Whereas the direction of vector x is described by unit vector a ˆn. Problem! There are two unit vectors that satisfy the equations ˆ a = ˆ n 0 and an = 0!! These two vectors are antiparallel. a ˆn 2 ˆ a = ˆ a = 0 n1 n2 ˆ a = ˆ a = 0 n1 n2 a ˆn 1 ˆ a = ˆ a n1 n2 Q: Which unit vector is correct? : Use the right-hand rule (See figure 2-9)!! From: class/1-right-hand-rule.html

21 8/19/2005 The Cross Product.doc 3/4 Some fun facts about the cross product! 1. If θ = 90 (i.e., orthogonal), then: = ˆ an = ˆ a n sin90 2. If θ = 0 (i.e., parallel), then: = ˆ a n = 0 sin0 Note that x =0 also if θ = The cross product is not commutative! In other words,. Instead: x I see! When evaluating the cross product of two vectors, the order is doggone important! x = ( ) 4. The negative of the cross product is: ( ) = = ( )

22 8/19/2005 The Cross Product.doc 4/4 5. The cross product is also not associative: ( ) C ( C ) Therefore, xxc has ambiguous meaning! 6. ut, the cross product is distributive, in that: and also, ( + C) = ( ) + ( C ) ( + C) = ( ) + ( C )

23 8/19/2005 The Triple Product.doc 1/3 The Triple Product The triple product is not a new operation, as it is simply a combination of the dot and cross products. The triple product of vectors,, and C is denoted as: xc Q: Yikes! Does this mean: or ( ) x C ( xc ) : The answer is easy! Only one of these two interpretations makes sense:

24 8/19/2005 The Triple Product.doc 2/3 In the first case, is a scalar value, say d =. Therefore we can write the first equation as: ( )xc= d xc ut, this makes no sense! The cross product of a scalar and a vector has no meaning. In the second interpretation, the cross product x C is a vector, say x C = D. Therefore, we can write the second equation as: ( x ) C = D Not only does this make sense, but the result is a scalar! The triple product xc results in a scalar value. The Cyclic Property It can be shown that the triple product of vectors,, and C can be evaluated in three ways: C x = C x = C x

25 8/19/2005 The Triple Product.doc 3/3 ut, it is important to note that this does not mean that order is unimportant! For example: xc Cx xc C x xc xc The cyclical rule means that the triple product is invariant to shifts (i.e., rotations) in the order of the vectors. C There are six ways to arrange three vectors. Therefore, we can group the triple product of three vectors into two groups of three products: C x = C x = C x xc = C x = Cx ( C) but, C x = x

26 8/22/2005 Example_Vector lgebra.doc 1/3 Example: Vector lgebra Consider the scalar expression: ac + bc + bd + ad We can manipulate and simply this expression using the rules of scalar algebra: ac + bc + bd + ad = ac + ad + bc + bd ( ac ad ) ( bc bd ) a ( c d ) b( c d ) ( a b)( c d ) = = = + + (commutative) (associative) (distributive) (distributive) We can likewise perform a similar analysis on vector expressions! Consider now the expression: ( + ) C We can show that this is actually a very familiar and basic vector operation! ( + ) C = C ( + ) (Triple product identity) ( ) = C + (Cross Product Distibutive) = C (Since = 0) = C (Triple product identity)

27 8/22/2005 Example_Vector lgebra.doc 2/3 Or, for example, if we consider: we find: ( + ) ( +2 ) ( ) ( 2) ( ) ( ) = + + = (dot product distributive) = (dot product distributive) = (scalar multiply commutative) = ( C C = C identity) = ( ) (dot product communitive) = (vector addition commutative) = (scalar multiply distributive) = (2+1=3) Keep in mind one very important point when doing vector algebra the expression can never change type (e.g., from vector to scalar)! In other words, if the expression initially results in a vector (or scalar), then after each manipulation, the result must also be a vector (or scalar).

28 8/22/2005 Example_Vector lgebra.doc 3/3 For example, we find that the following expression cannot possibly be true! + ( C) = ( C) + ( + ) C Q: Do you see why? : Likewise, be careful not to create expressions that have no mathematical meaning whatsoever! Examples include: ( ) C ( C ) +

29 8/22/2005 Scalar Vector or Neither.doc 1/1 Scalar, Vector, or Neither? Let s test our vector algebraic skills! Can you evaluate the following expressions, and determine whether the result is a scalar (S), a vector (V), or neither (N)?? 1. ( ) C 2. + ( C ) 3. ( ) C 4. ( xc ) 5. ( C) C( ) 6. ( x ) + ( + ) C C 7. xc D

30 8/22/2005 Orthogonal and Orthonormal Vector Sets.doc 1/4 Orthogonal and Orthonormal Vector Sets We often specify or relate a set of scalar values (e.g., x, y, z) using a set of scalar equations. For example, we might say: x = y and z = x + 2 From which we can conclude a third expression: z = y + 2 Say that we now add a new constraint to the first two: x + y = 2 We can now specifically conclude that: x = 1 y = 1 z = 3 Note we can likewise use vector equations to specify or relate a set of vectors (e.g., C).,, For example, consider a set of three vectors that are oriented such that they are mutually orthogonal!

31 8/22/2005 Orthogonal and Orthonormal Vector Sets.doc 2/4 In other words, each vector is perpendicular to each of the other two: C Note that we can describe this orthogonal relationship mathematically using three simple equations: Α = Α C = C = We can therefore define an orthogonal set of vectors using the dot product: Three (non-zero) vectors, and C form an orthogonal set iff they satisfy = C = C = 0

32 8/22/2005 Orthogonal and Orthonormal Vector Sets.doc 3/4 Note that there are an infinite number of mutually orthogonal vector sets that can be formed! E.G: Q: Can four or more vectors form an mutually orthogonal set? : Nope, not in a threedimensional space! Three is the limit. Consider now a mutually orthogonal set of unit vectors. Such a set can be defined as any three vectors that satisfy these six equations: = C = C = 0 (mutually orthognal) = = CC = 1 (unit magnitudes)

33 8/22/2005 Orthogonal and Orthonormal Vector Sets.doc 4/4 set of vectors that satisfy these equations are said to form an orthonormal set of vectors! Therefore, an orthonormal set consists of unit vectors where: ˆ a ˆ a = ˆ a ˆ a = ˆ a ˆ a = 0 C C gain, there are an infinite number of orthonormal vector sets, but each set consists of only three vectors.