Vector Algebra: Angles, Products, and Applications

Practice vocabulary flashcards covering vector operations, dot products, cross products, and their geometric applications from the test review notes.

Angle Between Two Vectors

The angle $\varphi$ between two vectors $\bar{a} = (a_1, a_2, a_3)$ and $\bar{b} = (b_1, b_2, b_3)$, computed using the inverse cosine of their dot product divided by the product of their magnitudes: $\varphi = \arccos\left(\frac{\bar{a} \cdot \bar{b}}{|\bar{a}| |\bar{b}|}\right)$.

Cosine of the Angle Between Vectors

The cosine of the angle $\alpha$ between vectors $\bar{a} = (a_1, a_2, a_3)$ and $\bar{b} = (b_1, b_2, b_3)$, given by the ratio of their dot product to the product of their magnitudes: $\cos(\alpha) = \frac{\bar{a} \cdot \bar{b}}{|\bar{a}| |\bar{b}|}$.

Vector Product (Cross Product)

An operation on two three-dimensional vectors $\bar{a} = (a_1, a_2, a_3)$ and $\bar{b} = (b_1, b_2, b_3)$ evaluated using the formal determinant: $\bar{a} \times \bar{b} = \begin{vmatrix} \bar{i} & \bar{j} & \bar{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$.

Anticommutativity of the Cross Product

A fundamental algebraic property of the vector cross product where reversing the order of the vector operands yields the opposite vector: $\bar{a} \times \bar{b} = -(\bar{b} \times \bar{a})$.

Distributivity of the Cross Product

The property that the vector cross product distributes across vector addition: $\bar{a} \times (\bar{b} + \bar{c}) = \bar{a} \times \bar{b} + \bar{a} \times \bar{c}$ and $(\bar{a} + \bar{b}) \times \bar{c} = \bar{a} \times \bar{c} + \bar{b} \times \bar{c}$.

Scalar Product (Dot Product)

An algebraic operation between two vectors $\bar{a} = (a_1, a_2, a_3)$ and $\bar{b} = (b_1, b_2, b_3)$ calculated by summing the products of their corresponding components: $\bar{a} \cdot \bar{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$.

Cross Product Applications: Geometric Areas

Geometric uses of the cross product magnitude to find the area of a parallelogram spanned by two vectors, $T_{\square} = |\bar{a} \times \bar{b}|$, and the area of a triangle, $T_{\triangle} = \frac{|\bar{a} \times \bar{b}|}{2}$.

Orthogonality Condition of Vectors

An application of the scalar dot product establishing that two vectors $\bar{a}$ and $\bar{b}$ are perpendicular if and only if their scalar product is zero: $\bar{a} \perp \bar{b} \iff \bar{a} \cdot \bar{b} = 0$.

Area of a Parallelogram Spanned by Vectors

The surface area of a parallelogram spanned by vectors $\bar{a} = (a_1, a_2, a_3)$ and $\bar{b} = (b_1, b_2, b_3)$, calculated as the magnitude of their cross product: $T_{\square} = |\bar{a} \times \bar{b}|$.