StudySmarter AI is coming soon!

- :00Days
- :00Hours
- :00Mins
- 00Seconds

A new era for learning is coming soonSign up for free

Suggested languages for you:

Americas

Europe

Q10.

Expert-verifiedFound in: Page 68

Book edition
7th

Author(s)
Douglas C. Giancoli

Pages
978 pages

ISBN
978-0321625922

**Given the vectors $\overrightarrow{A}$** **and $\overrightarrow{B}$** **in figure below, determine $\overrightarrow{B}-\overrightarrow{A}$****. (b) Determine $\overrightarrow{A}-\overrightarrow{B}$**** without using your answer in (a). Then compare your results your results and see if they are opposite.**

(a) The X and Y components of vector $\overrightarrow{B}-\overrightarrow{A}$ are -54.57 and 2.68, respectively.

(b) The X and Y components of vector $\overrightarrow{A}-\overrightarrow{B}$ are 54.57 and -2.68, respectively.

**Vector addition can be done by resolving components of individual vectors along the axes and adding corresponding components.**

The magnitude of vector $\overrightarrow{A}$, *A* = 44.0

The magnitude of vector $\overrightarrow{B}$, *B* = 26.5

**Resolving components of vector $\overrightarrow{A}$**

X-component of vector $\overrightarrow{A}$:

$\begin{array}{c}{A}_{\mathrm{x}}=A\mathrm{cos}26\xb0\\ =44.0\times \mathrm{cos}26\xb0\\ =39.55\end{array}$

Y-component of vector $\overrightarrow{A}$:

$\begin{array}{c}{A}_{\mathrm{y}}=A\mathrm{cos}64\xb0\\ =44.0\times \mathrm{cos}64\xb0\\ =19.29\end{array}$

**Resolving components of vector $\overrightarrow{B}$**

X-component of vector $\overrightarrow{B}$:

$\begin{array}{c}{B}_{\mathrm{x}}=-B\mathrm{cos}26\xb0\\ =-26.5\times \mathrm{cos}56\xb0\\ =-14.82\end{array}$

Y-component of vector $\overrightarrow{B}$:

$\begin{array}{c}{B}_{\mathrm{y}}=B\mathrm{cos}34\xb0\\ =26.5\times \mathrm{cos}34\xb0\\ =21.97\end{array}$

** **

The component of $\overrightarrow{B}-\overrightarrow{A}$ along the x-direction is given as:

$\begin{array}{c}{\left(\overrightarrow{B}-\overrightarrow{A}\right)}_{\mathrm{x}}={B}_{\mathrm{x}}-{A}_{\mathrm{x}}\\ =-14.82-39.55\\ =-54.57\end{array}$

The component of $\overrightarrow{B}-\overrightarrow{A}$ along the y-direction is given as:

$\begin{array}{c}{\left(\overrightarrow{B}-\overrightarrow{A}\right)}_{\mathrm{y}}={B}_{\mathrm{y}}-{A}_{\mathrm{y}}\\ =21.97-19.29\\ =2.68\end{array}$

$\overrightarrow{B}-\overrightarrow{A}$ vector has components -54.57 and 2.68 in the x and y-direction, respectively.

The component of $\overrightarrow{A}-\overrightarrow{B}$ along the x-direction is given as:

$\begin{array}{c}{\left(\overrightarrow{A}-\overrightarrow{B}\right)}_{\mathrm{x}}={A}_{\mathrm{x}}-{B}_{\mathrm{x}}\\ =39.55+14.82\\ =54.57\end{array}$

The component of $\overrightarrow{A}-\overrightarrow{B}$ along the y-direction is given as:

$\begin{array}{c}{\left(\overrightarrow{A}-\overrightarrow{B}\right)}_{\mathrm{y}}={A}_{\mathrm{y}}-{B}_{\mathrm{y}}\\ =19.29-21.97\\ =-2.68\end{array}$

$\overrightarrow{A}-\overrightarrow{B}$ vector has components 54.57 and -2.68 in the x and y-direction, respectively.

As the values of ${\left(\overrightarrow{A}-\overrightarrow{B}\right)}_{\mathrm{x}}=-{\left(\overrightarrow{B}-\overrightarrow{A}\right)}_{\mathrm{x}}$ and ${\left(\overrightarrow{A}-\overrightarrow{B}\right)}_{\mathrm{x}}=-{\left(\overrightarrow{B}-\overrightarrow{A}\right)}_{\mathrm{x}}$; thus, we can say that vectors $\overrightarrow{A}-\overrightarrow{B}$ and $\overrightarrow{B}-\overrightarrow{A}$ are equal in magnitude but opposite in direction.

94% of StudySmarter users get better grades.

Sign up for free