Cross product - vector, Mathematics

Cross Product

In this last section we will look at the cross product of two vectors.  We must note that the cross product needs both of the vectors to be three dimensional (3D) vectors.  

 As well, before getting into how to calculate these we should point out a major variation in between dot products and cross products. The product of a dot product is a number and the result of a cross product is a vector!  Be cautious not to confuse the two.

Thus, let's begin with the two vectors a = (a1, a2, a3) illustrated by the formula, and b = (b1, b2 , b3) then the cross product is illustrated by formula

a * b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)

This is not a simple formula to remember.  There are two methods to derive this formula.  Both of them make use of the fact that the cross product is actually the determinant of a 3x3 matrix.  If you don't be familiar with what this is that is don't worry about it.  You don't require to know anything about matrices or determinants to make use of either of the methods.  The notation for the determinant is like this,

473_Cross Product - Vector 3.png

The first row in the above determinant is the standard basis vectors and should appear in the order given here.  The 2nd row is the components of a? and the third row is the components of b.  Now, let's take a look at the dissimilar methods for getting the formula.

 The first technique uses the Method of Cofactors.  If you do not know the method or technique of cofactors that is fine, the result is all that we want.  Formula is given below:

103_Cross Product - Vector 2.png

This formula is not as hard to remember as it might at first come out to be.  First, the terms change in sign and notice that the 2x2 is missing the column below the standard basis vector that multiplies it also the row of standard basis vectors.

The second method is little easier; though, many textbooks don't cover this method as it will only work on 3x3 determinants.  This technique says to take the determinant as listed above and after that copy the first two columns onto the end as displayed below.

2002_Cross Product - Vector 1.png

We now have three diagonals which move from left to right and three diagonals which move from right to left.  We multiply all along each diagonal and add those that move from left to right and subtract those which move from right to left.

Posted Date: 4/13/2013 2:43:26 AM | Location : United States

Related Discussions:- Cross product - vector, Assignment Help, Ask Question on Cross product - vector, Get Answer, Expert's Help, Cross product - vector Discussions

Write discussion on Cross product - vector
Your posts are moderated
Related Questions
Proof of the Properties of vector arithmetic Proof of a(v → + w → ) = av → + aw → We will begin with the two vectors, v → = (v 1 , v 2 ,..., v n )and w? = w

The vertices of a ? ABC are A(4, 6), B(1. 5) and C(7, 2). A line is drawn to intersect sides AB and AC at D and E respectively, such that AD/AB = AE/AC = 1/4 .Calculate the  ar

In 6th grade I am learning about ratios rates and fractions. I am working on and need serious.

Calculate Moving Average The table given below represents company sales; calculate 3 and 6 monthly moving averages, for data Months Sales

Mrs. Farrell's class has 26 students. Just 21 were present on Monday. How many were absent? Subtract the number of students present from the total number within the class to de

what are the applications of de moiver''s theorem in programming and software engineering

How to Converting Percents to Fractions ? To convert a percent to a fraction: 1. Remove the percent sign. 2. Create a fraction, in which the resulting number from Step 1 is

The subsequent force that we want to consider is damping. This force may or may not be there for any specified problem. Dampers work to counteract any movement. There are some w

cot functions