8.4

The Dot Product of Two Vectors

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

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

• 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. $W = F ⋅ \overrightarrow{PQ}$