8.4

The Dot Product of Two Vectors

  • The dot product of u=⟨𝑢<em>1,𝑢</em>2⟩u = ⟨𝑢<em>1, 𝑢</em>2⟩ and v=⟨𝑣<em>1,𝑣</em>2⟩v = ⟨𝑣<em>1, 𝑣</em>2⟩ is a scalar value.

  • Formula: u⋅v=u<em>1v</em>1+u<em>2v</em>2u ⋅ v = u<em>1v</em>1 + u<em>2v</em>2

  • Properties of the Dot Product:

    1. u⋅v=v⋅uu ⋅ v = v ⋅ u

    2. u⋅(v+w)=u⋅v+u⋅wu ⋅ (v + w) = u ⋅ v + u ⋅ w

    3. c(u⋅v)=(cu)⋅v=u⋅(cv)c(u ⋅ v) = (cu) ⋅ v = u ⋅ (cv)

    4. 0⋅v=00 ⋅ v = 0

    5. v⋅v=∣∣v∣∣2v ⋅ v = ||v||^2

The Angle Between Two Vectors

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

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

  • Two vectors uu and vv are orthogonal if and only if u⋅v=0u ⋅ 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 WW done by a constant force FF:

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

    2. W=F⋅PQ→W = F ⋅ \overrightarrow{PQ}