1. Vectors — formulas and resources

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|\vec A|=\sqrt{A_x^2+A_y^2+A_z^2}

computes the magnitude of a vector from its Cartesian components.

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\hat u_A=\frac{\vec A}{|\vec A|}

produces a vector of magnitude one with the same direction and sense as $\vec A$ .

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\vec A=A_x\hat i+A_y\hat j+A_z\hat k

In two dimensions

A_x=A\cos\theta,\qquad A_y=A\sin\theta

Expresses a vector through its Cartesian components; in the plane, those components are obtained from the magnitude and the angle.

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\vec R=\sum_i\vec A_i

By components

R_x=\sum_i A_{ix},\qquad R_y=\sum_i A_{iy},\qquad R_z=\sum_i A_{iz}

The resultant vector is the sum of the vectors, equivalent to adding their components.

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\vec A\cdot\vec B=AB\cos\theta

Also

\vec A\cdot\vec B=A_xB_x+A_yB_y+A_zB_z

its result is a scalar.

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\vec A\times\vec B=AB\sin\theta\,\hat n

the result is perpendicular to the plane formed by $\vec A$ and $\vec B$ .