Unit vectors are perpendicular. Their dot product is zero. That means that no unit vector has any component that is parallel to another unit vector.
The term collinear is used to describe vectors which are scalar multiples of one another (they are parallel; can have different magnitudes in the same or opposite direction). The term coplanar is used to describe vectors in at least 3-space. Coplanar vectors are three or more vectors that lie in the same plane (any 2-D flat surface).
That fact alone doesn't tell you much about the original two vectors. It only says that (magnitude of vector-#1) times (magnitude of vector-#2) times (cosine of the angle between them) = 1. You still don't know the magnitude of either vector, or the angle between them.
Dot Product:Given two vectors, a and b, their dot product, represented as a ● b, is equal to their magnitudes multiplied by the cosine of the angle between them, θ, and is a scalar value.a ● b = ║a║║b║cos(θ)Cross Product:Given two vectors, a and b, their cross product, which is a vector, is represented as a X b and is equal to their magnitudes multiplied by the sine of the angle between them, θ, and then multiplied by a unit vector, n, which points perpendicularly away, via the right-hand rule, from the plane that a and b define.a X b = ║a║║b║sin(θ)n
If these are vectors, then ba = - ab
Given two vectors a and b, the area of a parallelogram formed by these vectors is:a x b = a*b * sin(theta) where theta is the angle between a and b, and where x is the norm/length/magnitude of vector x.
No.
They are unit vectors in the positive directions of the x and y axes.
A vector rotation in math is done on a coordinate plane.2D vectors can be rotated using the cross and dot product.3D vectors are rotated using matrix based quaternion math.
Yes. This is the basis of cartesian vector notation. With cartesian coordinates, vectors in 2D are represented by two vectors, those in 3D are represented by three. Vectors are generally represented by three vectors, but even if the vector was not in an axial plane, it would be possible to represent the vector as the sum of two vectors at right angles to eachother.
Yes., and their being along the coordinate axes does not change the answer.Consider the vectors: i, -i and j where i is the unit vector along the x axis and j along the y axis. The resultant of the three is j.
The cosine of the angle between two vectors is used in the dot product because it measures the similarity or alignment of the vectors. The dot product calculates the product of the magnitudes of the vectors and the cosine of the angle between them, resulting in a scalar value that represents the degree of alignment or correlation between the vectors.
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.
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.
For two vectors A and B, the scalar product is A.B= -ABcos(AB), the minus sign indicates the vectors are in the same direction when angle (AB)=0; the vector product is ABsin(AB). Vectors have the rule: i^2= j^2=k^2 = ijk= -1.
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.
That really depends on the type of vectors. Operations on regular vectors in three-dimensional space include addition, subtraction, scalar product, dot product, cross product.
All vectors that are perpendicular (their dot product is zero) are orthogonal vectors.Orthonormal vectors are orthogonal unit vectors. Vectors are only orthonormal if they are both perpendicular have have a length of 1.