Two unit vectors, a 1 and a 2 , lie in the x y plane and pass through the origin. They make angles 1

Two unit vectors, a₁ and a₂, lie in the x_y plane and pass through the origin. They make angles ϕ₁ and ϕ₂, respectively, with the x axis

(a) Express each vector in rectangular components;

(b) Take the dot product and verify the trigonometric identity, cos(ϕ₁ − ϕ₂) = cos ϕ₁ cos ϕ₂ + sin ϕ₁ sin ϕ₂;

(c) Take the cross product and verify the trigonometric identity sin(ϕ₂ − ϕ₁) = sin ϕ₂ cos ϕ₁ − cos ϕ₂ sin ϕ₁.