Vectors in 2D and 3D

A vector is a quantity possessing both magnitude and direction, applied in physics for velocity, force, and displacement.

Expressed via standard basis vectors $\hat{i}$, $\hat{j}$, $\hat{k}$:

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

Common pitfall: Vectors add tip-to-tail, not by adding scalar lengths. Walking 3 km east then 4 km north yields a 5 km displacement, not 7 km. Magnitudes only add when directions align.

Vector Properties

Vector operations obey strict algebraic laws essential for multidimensional problem solving.

Worked Example: For $\vec{v} = (3, 4, 0)$, magnitude is $\|\vec{v}\| = \sqrt{3^2 + 4^2 + 0^2} = 5$. Its unit vector is $\hat{v} = (3/5, 4/5, 0)$.

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