Vector Algebra: Unit Vectors, Cross Products, and Triangle Area

Vocabulary flashcards covering vector operations, cross product calculations, unit normal vectors, position vectors, and vector applications in finding the area of a triangle.

Unit Vector Perpendicular to Two Vectors

A vector of unit length that is orthogonal to both vectors $\vec{c}$ and $\vec{d}$, calculated using the formula $\frac{\vec{c} \times \vec{d}}{|\vec{c} \times \vec{d}|}$.

Cross Product Determinant Form

The mathematical method used to calculate the vector product $\vec{c} \times \vec{d}$ by expanding a $3 \times 3$ determinant containing the unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$ in the first row and the respective scalar components of each vector in the remaining rows.

Position Vector Difference

The vector connecting an initial point $A$ to a terminal point $B$, determined by subtracting the position vector of $A$ from the position vector of $B$: $\vec{AB} = \text{P.V. of } B - \text{P.V. of } A$.

Vector Magnitude

The length or scalar size of a three-dimensional vector with components $x\hat{i} + y\hat{j} + z\hat{k}$, calculated as $\sqrt{x^2 + y^2 + z^2}$.

Area of a Triangle Using Vectors

The geometric measure of a triangle defined by vertices $A$, $B$, and $C$, calculated from two adjacent side vectors as $\frac{1}{2}|\vec{AB} \times \vec{AC}|$ (where the cross-product magnitude $|\vec{AB} \times \vec{AC}|$ yields $\sqrt{21}$ for points $A(1, 1, 1)$, $B(1, 2, 3)$, and $C(2, 3, 1)$).