Solved Problem on Vectors
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The components of a vector are vx=−8 and vy=6. What are the magnitude and direction of this vector?


Problem data:
  • Component of the vector in the x direction:    vx=−8;
  • Component of the vector in the y direction:    vy=6.
Solution:

The magnitude of the vector is given by (Figure 1)
\[ \begin{gather} \bbox[#99CCFF,10px] {v^2=v_x^2+v_y^2} \end{gather} \]
\[ \begin{gather} v^2=(-8)^2+6^2\\[5pt] v=\sqrt{64+36\;}\\[5pt] v=\sqrt{100\;} \end{gather} \]
\[ \begin{gather} \bbox[#FFCCCC,10px] {v=10} \end{gather} \]
Figure 1

The direction is given by (Figure 2)
\[ \begin{gather} \bbox[#99CCFF,10px] {\theta=\arctan\frac{v_y}{v_x}} \end{gather} \]
\[ \begin{gather} \theta=\arctan\frac{6}{-8} \end{gather} \]
since arctan is an odd function,   \( f(-x)=-f(x) \)
Figure 2

\[ \begin{gather} \theta=-\arctan\frac{6}{8}\\[5pt] \theta=-36,9° \end{gather} \]
since the value of vx​ is negative, we add 180°
\[ \begin{gather} \theta=-36,9°+180° \end{gather} \]
\[ \begin{gather} \bbox[#FFCCCC,10px] {\theta=143,1°} \end{gather} \]
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