Question

If $\stackrel{\mathbf{\rightharpoonup }}{{\mathbf{F}}_{\mathbf{1}}}\mathbf{=}<300,0,-200>$and , calculate the following

If and , calculate the following:(a) $$\begin{gathered} \stackrel{\mathbf{\rightharpoonup }}{{\mathbf{F}}_{\mathbf{1}}}\mathbf{+}\stackrel{\mathbf{\rightharpoonup }}{{\mathbf{F}}_{\mathbf{2}}}\mathbf{}\mathbf{\left(}\mathit{b}\mathbf{\right)}\mathbf{}\left|\stackrel{\rightharpoonup }{F_1}+\stackrel{\rightharpoonup }{F_2}\right|\mathbf{}\mathbf{\left(}\mathit{c}\mathbf{\right)}\mathbf{}\left|\stackrel{\rightharpoonup }{F_1}+\stackrel{\rightharpoonup }{F_2}\right|\mathbf{}\mathbf{\left(}\mathit{d}\mathbf{\right)}\mathbf{}\left|\stackrel{\rightharpoonup }{F_1}+\stackrel{\rightharpoonup }{F_2}\right|\mathbf{=}\left|\stackrel{\rightharpoonup }{F_1}\right|\mathbf{+}\left|\stackrel{\rightharpoonup }{F_2}\right|\mathbf{?}\mathbf{}\mathbf{\left(}\mathit{e}\mathbf{\right)}\mathbf{}{\stackrel{\mathbf{\rightharpoonup }}{\mathbf{F}}}_{\mathbf{1}}\mathbf{-}\stackrel{\mathbf{\rightharpoonup }}{{\mathbf{F}}_{\mathbf{2}}}\mathbf{} \\ \mathbf{\left(}\mathit{f}\mathbf{\right)}\mathbf{}\left|\stackrel{\rightharpoonup }{F_1}-\stackrel{\rightharpoonup }{F_2}\right|\mathbf{}\mathbf{\left(}\mathit{g}\mathbf{\right)}\mathbf{}\mathbf{}\left|\stackrel{\rightharpoonup }{F_1}\right|\mathbf{-}\left|\stackrel{\rightharpoonup }{F_2}\right|\mathbf{}\mathbf{}\mathbf{\left(}\mathit{h}\mathbf{\right)}\mathbf{}\mathit{I}\mathit{s}\mathbf{}\left|\stackrel{\rightharpoonup }{F_1}-\stackrel{\rightharpoonup }{F_2}\right|\mathbf{=}\left|\stackrel{\rightharpoonup }{F_1}\right|\mathbf{-}\left|\stackrel{\rightharpoonup }{F_2}\right|\mathbf{?} \end{gathered}$$

Short answer

The answer to each part is:

Step-by-step solution

Basic rule of addition or subtraction of vectors

To add or subtract two vectors, add or subtract the corresponding components. This process is just like adding two constant numbers. Only the parts having the same type of components need to be added or subtracted.

Finding the sum of given vectors

(a)

Add the corresponding components of both the force vectors to obtain the sum of the given vectors

Therefore, the vector for is .

Finding the magnitude of the sum of given vectors

(b)

Take the square root of the sum of the square of each component of the vector to obtain its magnitude.

$$\begin{gathered} \left|\stackrel{\rightharpoonup }{F_1}+\stackrel{\rightharpoonup }{F_2}\right|=\sqrt{{\left(450\right)}^2+{\left(-350\right)}^2+{\left(-200\right)}^2} \\ =\sqrt{202500+90000+40000} \\ =\sqrt{332500} \\ =576.6 \\ Therefore,thevalueof\left|\stackrel{\rightharpoonup }{F_1}+\stackrel{\rightharpoonup }{F_2}\right|is576.6 \end{gathered}$$

Finding the value of F1⇀+F2⇀

(c)

Take the square root of the sum of the square of each component of the vector to obtain its magnitude.

$$\begin{gathered} \left|\stackrel{\rightharpoonup }{F_1}\right|=\sqrt{{\left(300\right)}^2+0^2+{\left(-200\right)}^2} \\ =\sqrt{90000+0+40000} \\ =\sqrt{130000} \\ =360.5 \end{gathered}$$

(c)

Take the square root of the sum of the square of each component of the vector to obtain its magnitude.

$$\begin{gathered} \left|\stackrel{\rightharpoonup }{F_1}\right|=\sqrt{{\left(150\right)}^2+{\left(-300\right)}^2-0^2} \\ =\sqrt{22500+90000+0} \\ =\sqrt{112500} \\ =335.4 \end{gathered}$$

Take the sum of obtained magnitudes.

$$\begin{gathered} \left|\stackrel{\rightharpoonup }{F_1}\right|+\left|\stackrel{\rightharpoonup }{F_2}\right|=360.5+335.4 \\ =695.9 \\ Thereforethevalueof\left|\stackrel{\rightharpoonup }{F_1}\right|+\left|\stackrel{\rightharpoonup }{F_2}\right|is695.9 \end{gathered}$$

Analyzing F1⇀+F2⇀=F1⇀+F2⇀

(d)

From part (b)$$\begin{gathered} \left|\stackrel{\rightharpoonup }{F_1}+\stackrel{\rightharpoonup }{F_2}\right|=576.6andpart\left(c\right)\left|\stackrel{\rightharpoonup }{F_1}\right|+\left|\stackrel{\rightharpoonup }{F_2}\right|=695.9 \\ Thereforethevalueof\left|\stackrel{\rightharpoonup }{F_1}+\stackrel{\rightharpoonup }{F_2}\right|isnotequaltovalueof\left|\stackrel{\rightharpoonup }{F_1}\right|+\left|\stackrel{\rightharpoonup }{F_2}\right|. \end{gathered}$$

Finding the difference between given vectors

(e)

Subtract the corresponding components of both the force vectors to obtain the difference between the given vectors.

Finding the magnitude of the difference between given vectors

(f)

Take the square root of the sum of the square of each component of the vector to obtain its magnitude.

$$\begin{gathered} \left|\stackrel{\rightharpoonup }{F_1}-\stackrel{\rightharpoonup }{F_2}\right|=\sqrt{{\left(150\right)}^2{\left(300\right)}^2+{\left(-200\right)}^2} \\ =\sqrt{22500+90000+40000} \\ =\sqrt{152500} \\ =390.5 \\ Thereforethevalueof\left|\stackrel{\rightharpoonup }{F_1}-\stackrel{\rightharpoonup }{F_2}\right|is390.5 \end{gathered}$$

Finding the value of F1⇀-F2⇀

(g)

From part (c), and $$\begin{gathered} \left|\stackrel{\rightharpoonup }{F_1}\right|=360.5and\left|\stackrel{\rightharpoonup }{F_2}\right|=335.4 \\ Takethedifferenceofobtainedmagnitudes. \\ \left|\stackrel{\rightharpoonup }{F_1}\right|-\left|\stackrel{\rightharpoonup }{F_2}\right|=360.5-335.4 \\ Thereforethevalueof\left|\stackrel{\rightharpoonup }{F_1}\right|-\left|\stackrel{\rightharpoonup }{F_2}\right|is25.1 \end{gathered}$$.

Analyzing F1⇀+F2⇀=F1⇀+F2⇀

(h)

From part (f)$$\begin{gathered} \left|\stackrel{\rightharpoonup }{F_1}-\stackrel{\rightharpoonup }{F_2}\right|=390.5andpartis\left(g\right)\left|\stackrel{\rightharpoonup }{F_1}\right|-\left|\stackrel{\rightharpoonup }{F_2}\right|=25.1 \\ Thereforethevalueof\left|\stackrel{\rightharpoonup }{F_1}\right|-\left|\stackrel{\rightharpoonup }{F_2}\right|isnotequaltothevalueof\left|\stackrel{\rightharpoonup }{F_1}\right|-\left|\stackrel{\rightharpoonup }{F_2}\right| \end{gathered}$$