Summarizing the Analytical Method of Vector Addition and Subtraction

Summary

• The analytical method of vector addition and subtraction involves using the Pythagorean theorem and trigonometric identities to determine the magnitude and direction of a resultant vector.
• The steps to add vectors \(\mathbf{A}\) and \(\mathbf{B}\) using the analytical method are as follows:

  Step 1: Determine the coordinate system for the vectors. Then, determine the horizontal and vertical components of each vector using the equations

  \({A}_{x} = A\text{ cos }\theta\)
  \({B}_{x} = B\text{ cos }\theta\)

  and

  \({A}_{y} = A\text{ sin }\theta\)
  \({B}_{x} = B\text{ sin }\theta\)

  Step 2: Add the horizontal and vertical components of each vector to determine the components \({R}_{x}\) and \({R}_{y}\) of the resultant vector, \(\mathbf{\text{R}}\):

  \({R}_{x}={A}_{x}+{B}_{x}\)

  and

  \({R}_{y}={A}_{y}+{B}_{y.}\)

  Step 3: Use the Pythagorean theorem to determine the magnitude, \(R\), of the resultant vector \(\mathbf{\text{R}}\):

  \(R=\sqrt{{R}_{x}^{2}+{R}_{y}^{2}}.\)

  Step 4: Use a trigonometric identity to determine the direction, \(\theta \), of \(\mathbf{\text{R}}\):

  \(\theta ={\text{tan}}^{-1}\left({R}_{y}/{R}_{x}\right).\)

Glossary

analytical method

the method of determining the magnitude and direction of a resultant vector using the Pythagorean theorem and trigonometric identities