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

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

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))