It is clear that the magnitude appears to be 1 after performing the necessary magnitude calculations on the determined unit vector. The unit vector of any vector v has the same direction as the vector v because unit vectors are only known to specify the direction of any vector v. In three dimensions, there is an additional axis known as the z-axis.

In three dimensions, the direction of unit vectors is important.

One component, the unit vector, is directed along the x-axis in a two-dimensional plane, while the other is directed along the y-axis. In two dimensions, the direction of unit vectors is important. In both two and three dimensions, the same formula is used to find unit vectors. Because the unit vector’s magnitude is already 1, it only accounts for the direction of the vector v.Ī vector must have a corresponding unit vector, whether it exists in a two-dimensional or three-dimensional plane. This is because the unit vector of any vector serves as the vector’s measuring unit. The vector v has a magnitude of 3, but it can also be represented in terms of the unit vector a, which has a magnitude of 1 but a north-east direction.Īny vector v’s unit vector must point in the same direction as the vector v. Any vector can become a unit vector when we divide it by the magnitude of the same given vector. Vector v = (1, 3) is not a unit vector, because its magnitude is not equal to 1, i.e., |v| = √(1+3) ≠ 1. This is why unit vectors are also referred to as ‘Direction Vectors.’ When unit vectors are used to scale a vector, the direction of that vector is revealed. Sal Khan was the one who came up with the idea.īecause unit vectors have a magnitude of 1, they only provide information about a vector’s direction. The unit vectors and can be combined to form any other vector. They are extremely beneficial for a variety of reasons. A unit vector is a vector with a magnitude of one. We can use the command TensorInnerProduct to check that the vectors X, Y, Z are indeed null vectors.A vector is a quantity that has both a magnitude and a direction associated with it. Y ≔ NullVector ⁡ σ, &phi 2 assuming a :: real, b :: real Use the command NullVector to find the corrresponding null vectors X, Y, Z. Use the command SolderForm to create a solder form &sigma. G := dt ⁢ dt − dx ⁢ dx − dy ⁢ dy − dz ⁢ dzĭefine an orthonormal tetrad F on M with respect to the metric g. G ≔ evalDG ⁡ dt &t dt − dx &t dx − dy &t dy − dz &t dz ĭGsetup ⁡ t, x, y, z, z1, z2, w1, w2, Mĭefine a spacetime metric g on M with signature 1, − 1, − 1, − 1. With ⁡ DifferentialGeometry : with ⁡ Tensor :įirst create the spinor bundle M with spacetime coordinates t, x, y, z and fiber coordinates z 1, z 2, w 1, w 2. It can always be used in the long form DifferentialGeometry:-Tensor:-NullVector. This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form NullVector(.) only after executing the commands with(DifferentialGeometry) with(Tensor) in that order.

With three arguments, the NullVector command returns the (complex) vector with components X i = &sigma iAA ' &phi A &psi A '.

With two arguments, the NullVector command returns the real vector with components X i = &sigma iAA ' &phi A &phi ‾ A ' The NullVector command accepts, as its first argument, a solder form with either covariant or contravariant tensor and spinor indices. Let &sigma be a solder form for the metric g, that is, &sigma is a rank 3 spin-tensor such that g ij = &sigma i AA ' &sigma jAA '. Let g be a metric on a 4-dimensional manifold with signature 1, − 1, − 1, − 1. &sigma - a spin-tensor defining a solder form on a 4-dimensional spacetime Tensor - construct a null vector from a solder form and a rank 1 spinor

Null vector example