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  2. Unit vector - Wikipedia

    en.wikipedia.org/wiki/Unit_vector

    When θ is a right angle, the versor is a right versor: its scalar part is zero and its vector part v is a unit vector in . Thus the right versors extend the notion of imaginary units found in the complex plane , where the right versors now range over the 2-sphere S 2 ⊂ R 3 ⊂ H {\displaystyle \mathbb {S} ^{2}\subset \mathbb {R} ^{3}\subset ...

  3. Euclidean vector - Wikipedia

    en.wikipedia.org/wiki/Euclidean_vector

    This matrix equation relates the scalar components of a in the n basis (u,v, and w) with those in the e basis (p, q, and r). Each matrix element c jk is the direction cosine relating n j to e k. [19] The term direction cosine refers to the cosine of the angle between two unit vectors, which is also equal to their dot product. [19] Therefore,

  4. Scalar projection - Wikipedia

    en.wikipedia.org/wiki/Scalar_projection

    Vector projection of a on b (a 1), and vector rejection of a from b (a 2). In mathematics, the scalar projection of a vector on (or onto) a vector , also known as the scalar resolute of in the direction of , is given by:

  5. Vector projection - Wikipedia

    en.wikipedia.org/wiki/Vector_projection

    The projection of a onto b can be decomposed into a direction and a scalar magnitude by writing it as = ^ where is a scalar, called the scalar projection of a onto b, and b̂ is the unit vector in the direction of b. The scalar projection is defined as [2] = ‖ ‖ ⁡ = ^ where the operator ⋅ denotes a dot product, ‖a‖ is the length of ...

  6. Vector field - Wikipedia

    en.wikipedia.org/wiki/Vector_field

    Given a subset S of R n, a vector field is represented by a vector-valued function V: S → R n in standard Cartesian coordinates (x 1, …, x n). If each component of V is continuous, then V is a continuous vector field. It is common to focus on smooth vector fields, meaning that each component is a smooth function (differentiable any number ...

  7. Vector (mathematics and physics) - Wikipedia

    en.wikipedia.org/wiki/Vector_(mathematics_and...

    Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w . In mathematics and physics , a vector space (also called a linear space) is a set whose elements, often called vectors , can be added together and multiplied ...

  8. Vector notation - Wikipedia

    en.wikipedia.org/wiki/Vector_notation

    where v x, v y, and v z are the scalar components of v. Scalar components may be positive or negative; the absolute value of a scalar component is its magnitude ...

  9. Curl (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Curl_(mathematics)

    If (x 1, x 2, x 3) are the Cartesian coordinates and (u 1, u 2, u 3) are the orthogonal coordinates, then = + + is the length of the coordinate vector corresponding to u i. The remaining two components of curl result from cyclic permutation of indices: 3,1,2 → 1,2,3 → 2,3,1.