Vectors in 3 Dimensions (Edexcel A-Level Mathematics): Flashcards

A 3D vector has components along $x$, $y$, and $z$ axes.

Sum corresponding components: $a + b = (a_x+b_x,...)$.

Magnitude: $|v| = \sqrt{v_x^2 + v_y^2 + v_z^2}$.

Dot product: $\mathbf{a} \cdot \mathbf{b} = a_x \cdot b_x + a_y \cdot b_y + a_z \cdot b_z$.

The cross product yields a vector perpendicular to both.

Use $\cos \theta = \frac{a \cdot b}{|a| |b|}$ to find $\theta$.

Area = $|a × b|$, the magnitude of the cross product.

Volume = $|a \cdot (b \times c)|$ using dot and cross products.

Unit vectors are $i$, $j$, $k$ along $x$, $y$, $z$ axes.

$v = (v_x, v_y, v_z) = v_x i + v_y j + v_z k$.