6.4 Vectors and dot Products
Dot Product
The dot product of u = ⟨u₁, u₂⟩ and v = ⟨v₁, v₂⟩ is given by
u ⋅ v = u₁v₁ + u₂v₂
Properties
Let u, v, and w be vectors in the plane or in space and let c be a scalar.
1. u ⋅ v = v ⋅ u
2. 0 ⋅ v = 0
3. u ⋅ (v + w) = u ⋅ w + u ⋅ v
4. v ⋅ v = ∥v∥²
5. c(u ⋅ v) = cu ⋅ v = u ⋅ cv
The dot product of u = ⟨u₁, u₂⟩ and v = ⟨v₁, v₂⟩ is given by
u ⋅ v = u₁v₁ + u₂v₂
Properties
Let u, v, and w be vectors in the plane or in space and let c be a scalar.
1. u ⋅ v = v ⋅ u
2. 0 ⋅ v = 0
3. u ⋅ (v + w) = u ⋅ w + u ⋅ v
4. v ⋅ v = ∥v∥²
5. c(u ⋅ v) = cu ⋅ v = u ⋅ cv
The Angle Between Two Vectors
The angle between two nonzero vectors is the angle θ, 0≤θ≤π, between their respective standard position vectors. It can be found using
cosθ = (u ⋅ v)/(∥u∥∥v∥)
The angle between two nonzero vectors is the angle θ, 0≤θ≤π, between their respective standard position vectors. It can be found using
cosθ = (u ⋅ v)/(∥u∥∥v∥)
The vectors u and v are orthogonal when u ⋅ v = 0
Orthogonal vectors meet at a 90 degree angle.
Orthogonal vectors meet at a 90 degree angle.
Finding Vector Components
Let u and v be nonzero vectors
u = w₁ + w₂
where w₁ and w₂ are orthogonal and w₁ is parallel to v. The vectors w₁ and w₂ are called vector components of u. The vector w₁ is the projection of u onto v and is denoted by
w₁ = projᵥu
The vector w₂ is given by
w₂ = u - w₁
To find the projection, the dot product can be used as
projᵥu = ((u ⋅ v)/(∥v∥²))v
Let u and v be nonzero vectors
u = w₁ + w₂
where w₁ and w₂ are orthogonal and w₁ is parallel to v. The vectors w₁ and w₂ are called vector components of u. The vector w₁ is the projection of u onto v and is denoted by
w₁ = projᵥu
The vector w₂ is given by
w₂ = u - w₁
To find the projection, the dot product can be used as
projᵥu = ((u ⋅ v)/(∥v∥²))v
Finding Force
Work
The work W done by a constant force F acting along the line of motion of an object is given by
W = (magnitude of force)(distance) = ∥F∥∥PQ∥
If the constant force F is not directed along the line of motion, then the work W done by the force is given by
W = ∥projPQF∥∥PQ∥
The work W done by a constant force F acting along the line of motion of an object is given by
W = (magnitude of force)(distance) = ∥F∥∥PQ∥
If the constant force F is not directed along the line of motion, then the work W done by the force is given by
W = ∥projPQF∥∥PQ∥