Question

Which of the following expressions best represents the dot product of two vectors? Select all that apply.
axbx + ayby
|a||b|(cosαcosβ + sinαsinβ)
|a||b|cos(α + β)
|a||b|cos(α - β)

Answer 1

Answer:

• axbx + ayby
• |a||b|(cosαcosβ + sinαsinβ)
• |a||b|cos(α - β)

Step-by-step explanation:

The dot product is the sum of products of the corresponding coordinate values:

$\text{\bf{a} \cdot \bf{b}}=a_{x}b_{x}+a_{y}b_{y}$

The value of this can also be written in terms of the magnitudes of the vectors and the angle θ between them:

  |a|·|b|·cos(θ)

But the angle between the vectors is the same as the difference of their individual angles, θ = α - β, so this can also be written as ...

  |a|·|b|·cos(α-β)

And the trig identity for the cosine of the difference of angles lets us write the above as ...

  = |a|·|b|·(cos(α)cos(β) +sin(α)sin(β))