8.4

The Dot Product of Two Vectors

• The dot product of u = ⟨𝑢1, 𝑢2⟩ and v = ⟨𝑣1, 𝑣2⟩ is a scalar value.

• Formula: u ⋅ v = u1v1 + u2v2

• Properties of the Dot Product:

  1. u ⋅ v = v ⋅ u

  2. u ⋅ (v + w) = u ⋅ v + u ⋅ w

  3. c(u ⋅ v) = (cu) ⋅ v = u ⋅ (cv)

  4. 0 ⋅ v = 0

  5. v ⋅ v = ||v||^2

The Angle Between Two Vectors

• The angle \theta between two nonzero vectors u and v can be determined from: cos \theta = \frac{u ⋅ v}{||u|| ||v||}.

• Alternative dot product calculation: u \cdot v = ||u|| ||v|| cos \theta

• Two vectors u and v are orthogonal if and only if u ⋅ v = 0.

Finding Vector Components

• Vectors w1w1 and w2w2 are vector components of uu.

• w1w1 is the projection of uu onto vv: projvu=u⋅v∣∣v∣∣2vprojvu=∣∣v∣∣2u⋅v​v

• w2w2 is given by: w2=u−projvuw2=u−projv​u

Work

• Work W done by a constant force F:

  1. W = ||F|| ||\overrightarrow{PQ}|| cos \theta, where \theta is the angle between the force and the displacement.

  2. W = F ⋅ \overrightarrow{PQ}