Evaluating with Vectors

Question

For the vectors u=(-3,-2) and v=(4,-2), evaluate the following:

 

a) The magnitude (length) of vector u (simplified radical form)

b) A unit vector in component form which has the same direction as v

c) The vector u+v (in a form using i's and j's)

d) The magnitude and direction angle of u+v

e) The dot product u•v

f) The angle between u and v

Answer 1

u = ⟨-3,-2⟩ and v = ⟨4,-2⟩

 

a) ||u|| = √[(-3)² + (-2)²] = √13

 

b) v/||v|| = ⟨4,-2⟩/√[4² + (-2)²] = ⟨4,-2⟩/√20 = ⟨4,-2⟩/(2√5) = ⟨2/√5 , -1/√5⟩

 

c) u+v = ⟨-3,-2⟩ + ⟨4,-2⟩ = ⟨1,-4⟩ = i - 4j

 

d) r = ||u+v|| = ||⟨1,-4⟩|| = √[1² + (-4)²] = √17

   

    θ = tan^-1 -4 ≈ -1.3258 radians (or -75.96°)

 

e) u•v = ⟨-3,-2⟩ • ⟨4,-2⟩ = (-3)(4) + (-2)(-2) = -8

 

f) cos θ = u•v / (||u|| ||v||) = -8/(√13 √20) = -4/√65

 

   θ = cos^-1 (-4/√65) ≈ 2.0899 radians (or 119.74°)