Vectors in 2D and 3D

A vector is a quantity with both magnitude and direction. In physics we use vectors for displacement, velocity, force, and many other quantities.

Notation and components

$$\vec{v} = v_x\,\hat{i} + v_y\,\hat{j} + v_z\,\hat{k}$$

Basic operations

• Addition: $\vec{u}+\vec{v} = (u_x+v_x,\; u_y+v_y,\; u_z+v_z)$
• Scalar mult.: $\alpha\vec{v} = (\alpha v_x,\; \alpha v_y,\; \alpha v_z)$
• Magnitude: $\|\vec{v}\| = \sqrt{v_x^{2} + v_y^{2} + v_z^{2}}$
• Unit vector: $\hat{v} = \vec{v}\,/\,\|\vec{v}\|$

Key properties

• Commutativity: $\vec{u}+\vec{v} = \vec{v}+\vec{u}$
• Associativity: $(\vec{u}+\vec{v})+\vec{w} = \vec{u}+(\vec{v}+\vec{w})$
• Zero vector: $\vec{v}+\vec{0} = \vec{v}$

Tip: Always sketch vectors before computing. A diagram catches sign errors that algebra misses.

Common pitfall: Vectors add tip-to-tail, not by adding lengths: walking 3 km east then 4 km north displaces you 5 km, not 7. Magnitudes only add when directions align.