Measure vector alignment and projection

Try

A displacement moves 3 m east and 4 m north. Represent it as vector u=(3,4) in an east–north frame. Its magnitude is 5 m, but its eastward component is only 3 m. Predict why multiplying its length by another vector’s length cannot describe directional alignment without considering their angle.

Predict

Compare u with an east-pointing vector v=(2,0), a west-pointing vector (−2,0) and a perpendicular vector (−4,3). Predict positive, negative and zero alignment respectively. A signed scalar can capture these differences even though every vector magnitude is nonnegative.

Experiment

Compute the dot product component by component.

Second vector v	u·v with u=(3,4)	Interpretation
(2,0)	6	Positive alignment
(−2,0)	−6	Opposing component
(−4,3)	0	Perpendicular directions

Draw all arrows from a common origin on equal-scale axes. The arrows can be translated without changing the represented displacement; their components are differences, not absolute point locations.

Understand

For u=(u₁,u₂) and v=(v₁,v₂), the Euclidean dot product is u·v=u₁v₁+u₂v₂. Magnitude is |u|=√(u·u). For nonzero vectors, u·v=|u||v|cos θ, where θ is the angle between them in [0,π]. Dividing by the magnitudes removes their length effects so that inverse cosine recovers direction difference.

The zero vector has no unique direction. It has dot product zero with every vector, but the angle formula cannot divide by its zero magnitude. State this condition whenever interpreting a zero dot product as a right angle between directions.

Projection onto a nonzero v is proj_v u=((u·v)/(v·v))v. For u=(3,4) and v=(2,0), the coefficient is 6/4=3/2 and the projected vector is (3,0). The coefficient is not itself the projected length unless the target vector has unit length. The signed scalar component along v is instead (u·v)/|v|.

For a constant force F and displacement d in the same frame, work is F·d. With F=(3,4) N and d=(2,−1) m, work is 2 joules. The negative vertical contribution subtracts from the positive horizontal contribution. A varying force requires additional integration rather than substituting one arbitrary force sample into this constant-force formula.

Apply

In practice, separate a scalar dot product, a vector projection and a magnitude. They answer different questions. Check units and frame consistency before interpreting results. In the angle task, justify why both vectors are nonzero and why the selected inverse-cosine angle lies in [0,π].

Continue

Polar coordinates describe a point through distance and direction rather than two component distances. The same circle relationships connect these representations, with extra care needed for angle nonuniqueness and negative radii.