Question

Let $\overrightarrow{\mathbf{A}}=60.0\text{cm}$at $270°$measured from the horizontal. Let data-custom-editor="chemistry" $\overrightarrow{\mathbf{B}}=80.0\text{cm}$at some angle $\theta$.

(a)Find the magnitude of$\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ as a function of.

(b) From the answer to part (a), for what value of$\theta$doesdata-custom-editor="chemistry" $|\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}|$take on its maximum value? What is this maximum value?

(c) From the answer to part (a), for what value of$\theta$does$|\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}|$take on its minimum value? What is this minimum value?

(d) Without reference to the answer to part (a), argue that the answers to each of parts (b) and (c) do or do not make sense.

Short answer

(a) The magnitude of $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$i s $\sqrt{\left(10000{\text{cm}}^2\right)-\left(9600{\text{cm}}^2\right)\sin \left(\theta \right)}$as a function of .

(b) For the value$\theta =270°$ ,the maximum value is$140\text{cm}$ .

(c) For the value$\theta =90°$ ,the maximum value is$20\text{cm}$ .

(d) Both the maximum and minimum value is correct

Step-by-step solution

Figure according to the question

According to the question,$\overrightarrow{\mathbf{A}}=60.0\text{cm}$ is at$270°$ and $\overrightarrow{\mathbf{B}}=80.0\text{cm}$at angle$\theta$ .

The magnitude of A→+B→

(a)

The addition of two vectores is expressed by R ,then the expression will be

$R=\sqrt{A^2+B^2+2AB\cos \varphi }$

In the above expression, A and B are the magnitude of two vectores and the angle is$\varphi$.

So,

localid="1663671219724" $$\begin{gathered} A=\left(60.0\text{cm}\right) \\ B=\left(80.0\text{cm}\right) \end{gathered}$$

And localid="1663671222921" $\varphi =\left(270°-\theta \right)$

So,the magnitude is:

The magnitude of$\overrightarrow{A}+\overrightarrow{B}$is $\sqrt{\left(10000{\text{cm}}^2\right)-\left(9600{\text{cm}}^2\right)\sin \left(\theta \right)}$as a function of$\theta$.

The maximum value of A→+B→

(b)

In the above equation,the term inside the square root will be maximum when the second term is minimum.In the second term,the minimum value is$-1$

$$\begin{gathered} \sin \theta =-1 \\ \Rightarrow \theta =si{n}^{-1}(-1)={270}^{°} \end{gathered}$$

So, $\overrightarrow{A}+\overrightarrow{B}$will be maximum when$\theta =270°$

After substitution the value,the maximum value is:

$$\begin{gathered} R=\sqrt{\left(10000{\text{cm}}^2\right)-\left(9600{\text{cm}}^2\right)\sin \left(\theta \right)} \\ =\sqrt{\left(10000{\text{cm}}^2\right)-\left(9600{\text{cm}}^2\right)\sin }\left(270°\right) \\ =140\text{cm} \end{gathered}$$

For the value$\theta =270°$ ,the maximum value is $140\text{cm}$.

The minimum value of A→+B→

(c)

In the above equation of step 2,the term inside the square root will be minimum when the $\theta$is minimum.So,the value of $\theta$is $-1$

$$\begin{gathered} \sin \theta =-1 \\ \Rightarrow \theta =si{n}^{-1}(-1)={90}^{°} \end{gathered}$$

So, $\overrightarrow{A}+\overrightarrow{B}$will be minimum when$\theta =90°$

After substitution the value,the minimum value is:

$$\begin{gathered} R=\sqrt{\left(10000{\text{cm}}^2\right)-\left(9600{\text{cm}}^2\right)\sin \left(\theta \right)} \\ =\sqrt{\left(10000{\text{cm}}^2\right)-\left(9600{\text{cm}}^2\right)\sin }\left(90°\right) \\ =20\text{cm} \end{gathered}$$

For the valuerole="math" localid="1663671041617" $\theta =90°$ ,the maximum value is$=20\text{cm}$ .

Posibility to get the answer without part (a)

(d)

When two vectors are parallel, the magnitude of their addition reaches its maximum. And the total magnitude is equal to the sum of the individual magnitudes. The magnitudes are$60\mathrm{cm}$ and$80\mathrm{cm}$ respectively. As a result,$\left|\overrightarrow{A}+\overrightarrow{B}\right|$ will have the maximum value of data-custom-editor="chemistry" $(60\text{cm})+(80\text{cm})=(140\text{cm})$, which is the same as the solution we found in part 1. (b). As a result, it is correct.

Similarly, the difference between the individual magnitudes will be the minimal magnitude of addition. As a result, $\left|\overrightarrow{A}+\overrightarrow{B}\right|$has a minimum value of$(80\text{cm})-(60\text{cm})=(20\text{cm})$ , which is the same as the solution we found in part 1. (c). As a result, it is correct.

So,both the maximum and minimum value is correct