To find the cross product of two vectors:
If a = [a1, a2, a3] and b = [b1, b2, b3], then
a x b = [a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1] or(a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)k
When the component vectors have equal or opposite directions (sin(Θ) = 0) i.e. the vectors are parallel.
(i) They are linearly dependent since the 2nd vector is twice the 1st vector. All 3 vectors lie in the x-z plane, so they don't span 3D space. (ii) They are linearly independent. Note that the cross-product of the first two is (-1,1,1). If the third vector is not perpendicular to the above cross-product, then the third vector does not lie in the plane defined by the first two vectors. (-1,1,1) "dot" (1,1,-1) = -1+1-1 = -1, not zero, so 3rd vector is not perpendicular to the cross product of the other two.
The products of the diagonal terms of two ratios is known as the cross product. This term is more often used in reference to vectors, however.
Perpendicular means that the angle between the two vectors is 90 degrees - a right angle. If you have the vectors as components, just take the dot product - if the dot product is zero, that means either that the vectors are perpendicular, or that one of the vectors has a magnitude of zero.
Multiply the product of their magnitudes by the cosine of the angle between them.
because that is the def. of a cross-product!
It depends on the type of product used. A dot or scalar product of two vectors will result in a scalar. A cross or vector product of two vectors will result in a vector.
Cross product is a mathematics term when there is a binary operation on two vectors in three-dimensional space.
It is the cross product of two vectors. The cross product of two vectors is always a pseudo-vector. This is related to the fact that A x B is not the same as B x A: in the case of the cross product, A x B = - (B x A).
Because there are two different ways of computing the product of two vectors, one of which yields a scalar quantity while the other yields a vector quantity.This isn't a "sometimes" thing: the dot product of two vectors is always scalar, while the cross product of two vectors is always a vector.
Cross product also known as vector product can best be described as a binary operation on two vectors in a three-dimensional space. The created vector is perpendicular to both of the multiplied vectors.
One type of cross is the cross or vector product of a pair of 3D vectors. If there are two unit vectors that are not parallel, their vector product is a vector that is normal to the plane containing the two vectors, so it's a good way to find that plane. In biological science, cross signifies the mating of two genotypes to produce its progeny. It may be among homozygous or heterozygous parents.
Normally you use sine theta with the cross product and cos theta with the vector product, so that the cross product of parallel vectors is zero while the dot product of vectors at right angles is zero.
If x is the angle between the two vectors then the magnitudes are equal if cos(x) = sin(x). That is, when x = pi/4 radians.
The cross product can be said to be a measure of the 'perpendicularity' of the vectors in the product. Please see the link.
A x B = |A| |B| sin[theta]
I think you meant to ask for finding a perpendicular vector, rather than parallel. If that is the case, the cross product of two non-parallel vectors will produce a vector which is perpendicular to both of them, unless they are parallel, which the cross product = 0. (a zero vector)