The Cross Product
We saw in the previous section on dot products that the dot product takes two vectors and produces a scalar, making it an example of a scalar product. In this section, we will introduce a vector product, a multiplication rule that takes two vectors and produces a new vector. We will find that this new operation, the cross product, is only valid for our 3-dimensional vectors, and cannot be defined in the 2- dimensional case.
One important feature of the dot product is its invariance under rotations.In other words, if we take a pair of vectors in the plane and rotate them both by the same angle, their dot product will remain the same. Consider the length of a single vector (which is given by the dot product): if the vector gets rotated about the origin by some angle, its length will not change--even though its direction can change quite dramatically! Similarly, from the geometric formula for the dot product, we see that the result depends only on the lengths of the two vectors and the angle between them. None of these quantities changes when we rotate the two vectors together, so neither can their dot product. This is what we mean when we say that the dot product is invariant under rotations.
Let's begin by defining the cross product for the unit vectors i , j , and k . Since all vectors can be decomposed in terms of unit vectors (see Unit vectors), once we've defined the cross products for this special case it will be easy to extend the definition to include all vectors. As we noted above, the cross product between iand j (since they both lie in the x - y plane) must point purely in the z -direction. Hence:
i×j = c k
for some constant c . Because later on we will want the magnitude of the resultant vector to have geometric significance, we need c k to have unit length. In other words, c can be either +1 or -1. Now we make a completely arbitrary choice in order to accord with convention: we choose c = + 1 . The fact that we have chosen c to be positive is known as The Right-Hand Rule.
Notice that the magnitude of the resultant vector is the same as the area of the rectangle with sides u and v ! As promised above, the magnitude of the cross product between two vectors, | u×v| , has a geometric interpretation. In general it is equal to the area of the parallelogram having the two given vectors as its sides (see ).
In particular, this means that for two parallel vectors the cross product equals 0.
where the resultant vector is perpendicular to each of the original two and its magnitude is given by | u×v| = | u|| v| sinθ .
We saw in the previous section on dot products that the dot product takes two vectors and produces a scalar, making it an example of a scalar product. In this section, we will introduce a vector product, a multiplication rule that takes two vectors and produces a new vector. We will find that this new operation, the cross product, is only valid for our 3-dimensional vectors, and cannot be defined in the 2- dimensional case.
One important feature of the dot product is its invariance under rotations.In other words, if we take a pair of vectors in the plane and rotate them both by the same angle, their dot product will remain the same. Consider the length of a single vector (which is given by the dot product): if the vector gets rotated about the origin by some angle, its length will not change--even though its direction can change quite dramatically! Similarly, from the geometric formula for the dot product, we see that the result depends only on the lengths of the two vectors and the angle between them. None of these quantities changes when we rotate the two vectors together, so neither can their dot product. This is what we mean when we say that the dot product is invariant under rotations.
Making the requirement of rotational invariance more stringent for the cross product, we need the cross product of two vectors to yield another vector. Consider, for instance, two 3-dimensional vectors u and v in a plane (two non-parallel vectors always define a plane, in the same way that two lines do. If we rotate this plane, the vectors will change direction, but we don't want the cross product w = u×v to change at all. However, if w has any non-zero components in the plane of u and v , those components will necessarily change under rotation (they get rotated just like everything else). The only vectors that won't change at all under a rotation of the u - v plane are those vectors that are perpendicular to the plane. Hence, the cross product of two vectors u and v must give a new vector which is perpendicular to both u and v .
we can see immediately that it is not possible to define a cross product for two- dimensional vectors, since there is no direction perpendicular to the plane of two-dimensional the vectors! (We'd need a third dimension for that).
Let's begin by defining the cross product for the unit vectors i , j , and k . Since all vectors can be decomposed in terms of unit vectors (see Unit vectors), once we've defined the cross products for this special case it will be easy to extend the definition to include all vectors. As we noted above, the cross product between iand j (since they both lie in the x - y plane) must point purely in the z -direction. Hence:
i×j = c k
for some constant c . Because later on we will want the magnitude of the resultant vector to have geometric significance, we need c k to have unit length. In other words, c can be either +1 or -1. Now we make a completely arbitrary choice in order to accord with convention: we choose c = + 1 . The fact that we have chosen c to be positive is known as The Right-Hand Rule.
It turns out that in order to be consistent with the Right-Hand Rule, all of the cross products between unit vectors are uniquely determined:
| i×j | = | k = - j×i | |
| j×k | = | i = - k×j | |
| k×i | = | j = - i×k |
In particular, notice that the order of the vectors within the cross products holds significance. In general, u×v = - v×u . From here we can see that the cross product of a vector with itself is always zero, since by the above rule u×u = - u×u , which means that both sides must vanish for equality to hold. We can now complete our list of cross products between unit vectors by observing that:
| i×i = j×j = k×k = 0 |
Geometric Formula for Cross Product
Consider the cross product of two (not necessarily unit-length) vectors that lie purely along the x and y axes (as i and j do). We can thus write the vectors as u = a i and v = b j , for some constants a and b . The cross product u×v is thus equal to
| u×v = ab(i×j) = ab k |
From basic geometry, we know that this area is given by area = | u|| v| sinθ , where| u| and | v| are the lengths of the sides of the parallelogram, and θ is the angle between the two vectors. Notice that when the two vectors are perpendicular to each other, θ = 90 degrees, so sinθ = 1 and we recover the familiar formula for the area of a square. On the other hand, when the two vectors are parallel, θ = 0 degrees, and sinθ =0, meaning the area vanishes (as we expect). In general, then, we find that the magnitude of the cross product between two vectors u and v that are separated by an angle θ (going clockwise from u to v , as specified by the Right-Hand Rule) is given by:
| | u×v| = | u|| v| sinθ |
Cross Product Summary
In summary, the cross product of two vectors is given by:
| u×v = (u 1 v 2 - u 2 v 1)k + (u 3 v 1 - u 1 v 3)j + (u 2 v 3 - u 3 v 2)i |
.
, where the fact that the inversion is m-modular is implicit.
corresponding to masses placed at the vertices of a reference triangle 
. These masses then determine a point 
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. The vertices of the triangle are given by 
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. Barycentric coordinates were discovered by Möbius in 1827 (Coxeter 1969, p. 217; Fauvel et al. 1993). 
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as the mass at 
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at 
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the centroid (left figure). Furthermore, the areas of the triangles 
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of 
(right figure; Coxeter 1969, p. 217). 
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