The dot or scalar product of two (3d) vec- tors ⃗a = ⟨a1,a2,a3⟩ and ⃗b = ⟨b1,b2,b3⟩ is defined as
3 ⃗a • ⃗b = a i b i = a 1 b 1 + a 2 b 2 + a 3 b 3 .
i=1
in 3d euclidean space, the scalar product has a geometric interpretation, given by ⃗a • ⃗b = ∥ ⃗a ∥ ∥ ⃗b ∥ c o s θ ,
where θ is the angle between ⃗a and ⃗b and the norm(length) of a vector ∥⃗a∥ is defined by
∥ ⃗a ∥ = √ ⃗a • ⃗a .
in words, the scalar product of two vectors can be thought of as the product of the magnitude of ⃗a with the magnitude of the projection of ⃗b onto the direction of ⃗a. it is used to calculate the product of vector quantities when only the parallel components of each vector contribute (e. g., work = force • displacement).
let ⃗a = ⟨14, 10.5, 0⟩ and ⃗b = ⟨4.62, 9.45, 0⟩. calculate ⃗a • ⃗b.