Vectors & 3D Geometry

A vector $\langle a, b, c \rangle$ encodes magnitude and direction; its length is $|\mathbf{v}| = \sqrt{a^2 + b^2 + c^2}$. The dot product $\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 = |\mathbf{u}||\mathbf{v}|\cos\theta$ measures alignment.

Two vectors are orthogonal exactly when their dot product is zero. The cross product $\mathbf{u} \times \mathbf{v}$ is perpendicular to both, with magnitude equal to the parallelogram area $|\mathbf{u}||\mathbf{v}|\sin\theta$ and direction from the right-hand rule.

Angle between $\langle 1, 1, 0 \rangle$ and $\langle 1, 0, 0 \rangle$: dot product $= 1$, lengths $\sqrt{2}$ and $1$, so $\cos\theta = \frac{1}{\sqrt{2}}$ and $\theta = 45^{\circ}$.