Математика22 мин

Vectors

2D vector arithmetic, magnitude, and the dot product — geometry without diagrams.

Components and magnitude

\vec{v} = (v_x, v_y)

Component form.

|\vec{v}| = \sqrt{v_x^2 + v_y^2}

\vec{u} + \vec{v} = (u_x + v_x,\ u_y + v_y)

k\vec{v} = (kv_x,\ kv_y)

Scalar multiplication scales each component.

\vec{u} \cdot \vec{v} = u_x v_x + u_y v_y = |\vec{u}||\vec{v}| \cos\theta

Perpendicular vectors: dot product is 0. Parallel vectors: components are proportional.

Find the angle

Angle between $\vec{u} = (1, 0)$ and $\vec{v} = (1, 1)$ ?

Жауап: 45°