Physics Practice Problems Vector Addition & Subtraction Practice Problems Solution: For the vectors shown in the figure, determine B -...

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Solution: For the vectors shown in the figure, determine B - 2A. Vector magnitudes are given in arbitrary units.

Problem

For the vectors shown in the figure, determine B - 2A. Vector magnitudes are given in arbitrary units.

Three vectors drawn on a rectangular coordinate system. Vector A has a magnitude of 44.0 units and makes an angle of 28.0 degrees with the positive x axis. Vector B has a magnitude of 26.5 units and makes an angle of 56.0 degrees with the negative x axis. Vector C has a magnitude of 31.0 units and lies along the negative y axis.

Solution

Whenever we're adding and subtracting vectors in 2D, we follow these steps:

  1. Draw a diagram and resolve the vectors into components.
  2. Organize your known information.
  3. Add or subtract vectors as required.
  4. Convert back to magnitude-angle notation (unless the problem only asks for components).

For step 1, we'll use these equations to find the x- and y-components of our given vectors:

Ax=|A| cos θAy=|A| sin θ

For step 4, we'll use these equations to calculate the magnitude and angle from the +x-axis:

|A|=Ax2+Ay2

tan θ=AyAx

Remember that if we solve the last equation for θ, we always have to check whether the result we get makes sense based on the components. We might need to add or subtract 180°, because the inverse tangent function only returns angles between −90° and 90°.

Step 1. For this problem, we're given a diagram, so we don't need to draw one from scratch. Unfortunately one of the given angles is measured clockwise from the negative x-axis, and we need  θB as a counterclockwise angle from the positive x-axis for our formulas to work:

θB=180°-56.0°=124°

Now we can plug in known information for B and get its components, keeping an extra significant figure to reduce rounding error:

Bx=B cos θB=(26.5) cos(124°)=-14.82  By=|B| sin θB=(26.5) sin(124°)=21.97

Since we're subtracting 2A, let's just go ahead and find 2Ax and 2Ay:

2Ax=|2A| cos θA=(88.0) cos(28.0°)=77.70  2Ay=|2A| sin θA=(88.0) sin(28.0°)=41.31

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