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In mathematics, a vector is simply an element of a vector space, and as such the central object of study of linear algebra. This is a general definition including a variety of mathematical objects (numbers, sequences, functions?, and operators). In physics and engineering the term vector usually refers to the particular vectors from Euclidean space (or from tangent spaces of a differentiable manifold), and represent quantities characterized by both magnitude? and direction. Examples are displacement?, velocity, momentum, force and acceleration. One also consideres bound or fixed vectors which are characterized by magnitude, direction and base point. Examples of these are position, torque and angular momentum. These two types of vectors are discussed below. [Vector fields]? can be thought off as rules that assign a bound vector to every point in space. Examples are the velocity field of a moving fluid or the magnetic? or [electrical field]?. |
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In mathematics, a vector is an element of a vector space, and as such the central object of study of linear algebra. This is a general definition including a variety of mathematical objects (numbers, sequences, functions, and operators). In physics and engineering the term vector usually refers to the particular vectors from Euclidean space (or from tangent spaces of a differentiable manifold), and represent quantities characterized by both magnitude? and direction. Examples are displacement?, velocity, momentum, force and acceleration. One also consideres bound or fixed vectors which are characterized by magnitude, direction and base point. Examples of these are position, torque and angular momentum. These two types of vectors are discussed below. [Vector fields]? can be thought off as rules that assign a bound vector to every point in space. Examples are the velocity field of a moving fluid or the magnetic or [electrical field]?. |
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* i×j = k, j×k = i, k×i = j * It follows that in [Cartesian coordinates]?, the cross product can be written as: :a×b = [a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1] * the above component notation can also be written as the determinant of the matrix: i j k a1 a2 a3 b1 b2 b3 |
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:a×b = [a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1] i j k a1 a2 a3 b1 b2 b3 |
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In seven dimensions it is also possible to define a cross product. This product has the following properties in common with the usual 3-dimensional cross product: * It is bilinear in the sense that x×(ay+bz)=ax×y+bx×z and (ay+bz)×x=ay×x+bz×x * It is anti-commutative: x×y=-y×x * x·(x×y)=y·(x×y)=0 * |x×y|2=|x|2|y|2(1-(x·y)2) and it satisfies the Jacobi identity above. Unfortunately the definition is more complex than that in three dimensions. The 3-D and 7-D cross products are related to the quaternions and octonions, respectively. |
In seven dimensions it is also possible to define a cross product. This product has the following properties in common with the usual 3-dimensional cross product:
and it satisfies the Jacobi identity above. Unfortunately the definition is more complex than that in three dimensions. The 3-D and 7-D cross products are related to the quaternions and octonions, respectively. |
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The scalar triple product (also called the box product or mixed triple product) isn't really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is denoted by (a b c) and defined as: |
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The scalar triple product (also called the box product or mixed triple product ) isn't really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is denoted by (a b c) and defined as: :(a b c) = a·(b×c) It has three primary uses. First, the absolute value of the box product is the volume of the parallelepiped which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors are linearly dependent , which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors a, b and c are oriented like the coordinate system i, j and k. In coordinates, if the three vectors are thought off as rows, the scalar triple product is simply the determinant of the 3-by-3 matrix having the three vectors as rows. The scalar triple product is linear in all three entries and anti-symmetric in the following sense: :(a b c) = (c a b) = (b c a) = -(a c b) = -(b a c) = -(c b a) |
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:(a b c) = a·(b×c) |
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It has three primary uses. First, the absolute value of the box product is the volume of the parallelepiped which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors are linearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors a, b and c are oriented like the coordinate system i, j and k. |
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In coordinates, if the three vectors are thought off as rows, the scalar triple product is simply the determinant of the 3-by-3 matrix having the three vectors as rows. The scalar triple product is linear in all three entries and anti-symmetric in the following sense: :(a b c) = (c a b) = (b c a) = -(a c b) = -(b a c) = -(c b a) |
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