Ch 01: Units & VectorsWorksheetSee all chapters
All Chapters
Ch 01: Units & Vectors
Ch 02: 1D Motion (Kinematics)
Ch 03: 2D Motion (Projectile Motion)
Ch 04: Intro to Forces (Dynamics)
Ch 05: Friction, Inclines, Systems
Ch 06: Centripetal Forces & Gravitation
Ch 07: Work & Energy
Ch 08: Conservation of Energy
Ch 09: Momentum & Impulse
Ch 10: Rotational Kinematics
Ch 11: Rotational Inertia & Energy
Ch 12: Torque & Rotational Dynamics
Ch 13: Rotational Equilibrium
Ch 14: Angular Momentum
Ch 15: Periodic Motion (NEW)
Ch 15: Periodic Motion (Oscillations)
Ch 16: Waves & Sound
Ch 17: Fluid Mechanics
Ch 18: Heat and Temperature
Ch 19: Kinetic Theory of Ideal Gasses
Ch 20: The First Law of Thermodynamics
Ch 21: The Second Law of Thermodynamics
Ch 22: Electric Force & Field; Gauss' Law
Ch 23: Electric Potential
Ch 24: Capacitors & Dielectrics
Ch 25: Resistors & DC Circuits
Ch 26: Magnetic Fields and Forces
Ch 27: Sources of Magnetic Field
Ch 28: Induction and Inductance
Ch 29: Alternating Current
Ch 30: Electromagnetic Waves
Ch 31: Geometric Optics
Ch 32: Wave Optics
Ch 34: Special Relativity
Ch 35: Particle-Wave Duality
Ch 36: Atomic Structure
Ch 37: Nuclear Physics
Ch 38: Quantum Mechanics

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

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


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.


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:


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:


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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