# Dictionary:Fig V-2

FIG. V-2. Vectors are quantities that have both magnitudes and directions, commonly represented by bold face, by arrows whose length is proportional to the vector’s magnitude, or by components. In Cartesian coordinates (with the unit orthogonal vectors i, j, and k),

${\displaystyle \mathbf {A} =a_{1}\mathbf {i} +a_{2}\mathbf {j} +a_{3}\mathbf {k} ;}$

${\displaystyle \mathbf {B} =b_{1}\mathbf {i} +b_{2}\mathbf {j} +b_{3}\mathbf {k} ;}$

${\displaystyle a_{i}=|\mathbf {A} |\cos(\mathbf {A} ,i)}$

${\displaystyle |\mathbf {A} |={\sqrt {{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}}}}$

${\displaystyle \mathbf {A} \pm \mathbf {B} =(a_{1}\pm b_{1})\mathbf {i} +(a_{2}\pm b_{2})\mathbf {j} +(a_{3}\pm b_{3})\mathbf {k} }$

Addition is shown in (a); the negative of a vector is represented by an arrow pointing in the opposite direction and subtraction by adding the negative vector.The dot product (or inner product) is not a vector but a scalar of magnitude

${\displaystyle \mathbf {A} \cdot \mathbf {B} =|\mathbf {A} ||\mathbf {B} |\cos(\mathbf {A} ,\mathbf {B} )=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3},}$

where the cosine is of the angle between their directions. The cross product (or outer product) is a vector perpendicular to both A and B in the direction a right-hand screw would advance if turned from A toward B (c, see also Figure I-3):

${\displaystyle \mathbf {A} \times \mathbf {B} =\mathbf {k|A||B|} \sin(\mathbf {A} ,\mathbf {B} )=(a_{2}b_{3}-a_{3}b_{2})\mathbf {i} +(a_{3}b_{1}-a_{1}b_{3})\mathbf {j} +a_{2}b_{1}-a_{1}b_{2})\mathbf {k} }$
${\displaystyle \mathbf {k} ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\end{vmatrix}}.}$

A vector field has values associated with every point in space (d).

Gradient: ${\displaystyle \nabla \mathbf {V} ={\frac {\partial }{\partial {x}}}\mathbf {i} +{\frac {\partial }{\partial {y}}}\mathbf {j} +{\frac {\partial }{\partial {z}}}\mathbf {k} ;}$
Divergence: ${\displaystyle \nabla \cdot \mathbf {V} ={\frac {\partial {V_{1}}}{\partial {x}}}+{\frac {\partial {V_{2}}}{\partial {y}}}+{\frac {\partial {V_{3}}}{\partial {z}}};}$
Curl: ${\displaystyle \nabla \times \mathbf {V} =({\frac {\partial {V_{3}}}{\partial {y}}}-{\frac {\partial {V_{2}}}{\partial {z}}})\mathbf {i} +({\frac {\partial {V_{1}}}{\partial {z}}}-{\frac {\partial {V_{3}}}{\partial {x}}})\mathbf {j} +({\frac {\partial {V_{2}}}{\partial {x}}}-{\frac {\partial {V_{1}}}{\partial {y}}})\mathbf {k} .}$

Vectors are not limited to three dimensions. Equivalent expressions in cylindrical and spherical coordinates are shown in Fig. C-14. For rotating vectors, see complex notation. (e) Vectors. (a) Addition (and subtraction) of vectors; (b) components of vectors and orthogonal unit vectors i, j, and k; (c) cross product of two vectors is orthogonal to both of them; (d) an increment to a vector is not necessarily in the direction of the vector; (e) vector operations.