Question

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.

Short answer

(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.

Step-by-step solution

Step 1. Addition of vectors

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

Step 2. Given data and assumptions

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

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

Step 3. Resolving vector components of A→ and B→ along the axes 

Resolving components of vector$\overrightarrow{A}$

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

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

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

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

Resolving components of vector$\overrightarrow{B}$

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

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

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

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

Step 4. Calculating the value of B→-A→

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.

Step 5. Calculating the value of A→-B→

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.

Step 6. Checking whether vectors B→-A→ and A→-B→ are equal and opposite or not

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.