Practice & Assess

Vectors: Addition, Subtraction & Projection

Drag vectors on the plane to explore u + v, u − v, and proj_v u with the dot product.

Vectors in the Plane

Add, subtract, and project vectors — with the dot product.

Addition: tip-to-tail / parallelogram.

u + v

Vector u = (3, 1)

u_x3

u_y1

Vector v = (1, 2.5)

v_x1

v_y2.5

∣ u ∣

3.16

∣ v ∣

2.69

u \cdot v

5.5

∠ (u, v)

49.8°

u + v

(4, 3.5)

Drag the tips of u and v on the plane.
In projection mode, the amber arrow is proj_v u; the dashed segment is the perpendicular from u to that line.

■ u■ v■ u + v

Green arrow is u + v. Dashed sides complete the parallelogram.

u \cdot v = u_{x} v_{x} + u_{y} v_{y} = 5.5

u + v

= (4, 3.5)