Vector has a magnitude of 122 units and points 37.0 ° north of west. Vector points 64.0 ° east of north. Vector points 17.0 ° west of south. These three vectors add to give a resultant vector that is zero. Using components, find the magnitudes of (a) vector and (b) vector .
To find the magnitudes of vectors A and B, we'll first break them down into their components using trigonometry.
Let's start with vector A, which points 37.0° north of west. To find its components, we'll use the following formulas:
Aᵢ = A * cos(θ)
Aⱼ = A * sin(θ)
where A is the magnitude of the vector and θ is the angle it makes with the x-axis (in this case, west).
Using vector A, we have:
Aᵢ = A * cos(37.0°)
Aⱼ = A * sin(37.0°)
Similarly, for vector B, which points 64.0° east of north, we have:
Bᵢ = B * sin(64.0°)
Bⱼ = B * cos(64.0°)
Finally, for vector C, which points 17.0° west of south:
Cᵢ = C * sin(17.0°)
Cⱼ = -C * cos(17.0°) [Note the negative sign because it is south]
Since the resultant vector is zero, the sum of the components of A, B, and C should also be zero:
(Aᵢ + Bᵢ + Cᵢ) = 0
(Aⱼ + Bⱼ + Cⱼ) = 0
Now, we can substitute the component formulas into these equations:
(A * cos(37.0°)) + (B * sin(64.0°)) + (C * sin(17.0°)) = 0
(A * sin(37.0°)) + (B * cos(64.0°)) - (C * cos(17.0°)) = 0
Given the magnitudes of A, B, and C, we can solve these two equations to find the unknowns, A and B.