Vectors and Operations on Them

Vector as a Geometric Object

A vector is a directed segment characterized by its length (magnitude) and direction. Two vectors are equal if they are parallel and of equal length—regardless of their initial point (free vectors).

Cartesian coordinates: a vector $\mathbf{a} = (a_1, a_2, a_3)$ is defined by its projections onto the axes. Magnitude $|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$.

Linear Operations

Addition: $\mathbf{a} + \mathbf{b} = (a_1 + b_1, a_2 + b_2, a_3 + b_3)$. Geometrically: parallelogram rule or triangle rule.

Multiplication by a scalar: $\lambda \mathbf{a} = (\lambda a_1, \lambda a_2, \lambda a_3)$. For $\lambda > 0$—direction is preserved; for $\lambda < 0$—direction is reversed.

Expansion in terms of basis: $\mathbf{a} = a_1\mathbf{e}_1 + a_2\mathbf{e}_2 + a_3\mathbf{e}_3$, where $\mathbf{e}_1$, $\mathbf{e}_2$, $\mathbf{e}_3$ are unit vectors along coordinate axes.

Scalar Product

$(\mathbf{a}, \mathbf{b}) = \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos \theta = a_1b_1 + a_2b_2 + a_3b_3$, where $\theta$ is the angle between the vectors.

Properties: symmetry, bilinearity, $(\mathbf{a}, \mathbf{a}) = |\mathbf{a}|^2 \geq 0$.

Orthogonality: $\mathbf{a} \perp \mathbf{b} \iff \mathbf{a} \cdot \mathbf{b} = 0$.

Projection of $\mathbf{b}$ on $\mathbf{a}$: $\mathrm{pr}_\mathbf{a} \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|}$.

Application: work $W = \mathbf{F} \cdot \mathbf{s} = |\mathbf{F}||\mathbf{s}|\cos \theta$. Physical applications of the scalar product are ubiquitous.

Vector Product

$\mathbf{a} \times \mathbf{b}$ is a vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$, of length $|\mathbf{a}||\mathbf{b}|\sin \theta$, directed according to the right-hand rule.

Computation via determinant: $ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3 \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ \end{vmatrix} = (a_2b_3 - a_3b_2)\mathbf{e}_1 - (a_1b_3 - a_3b_1)\mathbf{e}_2 + (a_1b_2 - a_2b_1)\mathbf{e}_3. $

Anticommutativity: $\mathbf{b} \times \mathbf{a} = -\mathbf{a} \times \mathbf{b}$.

Area of parallelogram: $S = |\mathbf{a} \times \mathbf{b}|$.

Application: moment of force $\mathbf{M} = \mathbf{r} \times \mathbf{F}$, magnetic force $\mathbf{F} = q\mathbf{v} \times \mathbf{B}$ (Lorentz formula).

Mixed Product

$(\mathbf{a}, \mathbf{b}, \mathbf{c}) = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \ \end{vmatrix}$.

Volume of parallelepiped: $V = |(\mathbf{a}, \mathbf{b}, \mathbf{c})|$.

Coplanarity: $(\mathbf{a}, \mathbf{b}, \mathbf{c}) = 0 \iff \mathbf{a}, \mathbf{b}, \mathbf{c}$ lie in one plane.

Property: $(\mathbf{a}, \mathbf{b}, \mathbf{c}) = (\mathbf{b}, \mathbf{c}, \mathbf{a}) = (\mathbf{c}, \mathbf{a}, \mathbf{b})$—invariant under cyclic permutations.