### Vector Magnitude Calculation#### (c) Use the projections of vector \(\overrightarrow{OP}\) along the x and y directions to calculate the magnitude of \(\overrightarrow{OP}\) using:\[ OP = \sqrt{(OP_x)^2 + (OP_y)^2} \]\[ OP = \_\_\_\_ \text{ cm} \]#### (d) Use the projections of vector \(\overrightarrow{OP}\) along the x' and y' directions to calculate the magnitude of \(\overrightarrow{OP}\) using:\[ OP = \sqrt{(OP_{x'})^2 + (OP_{y'})^2} \]\[ OP = \_\_\_\_ \text{ cm} \]#### (e) ConclusionWhat can you conclude about the magnitude of a vector with respect to rotated coordinate axes?**Conclusion:** The magnitude of the vectors using components in different coordinate axes that are rotated with respect to each other remains unchanged. The image presents a vector problem involving two sets of perpendicular axes. Here's the transcription:---**1.**The diagram displays a vector \(\overrightarrow{OP}\) with a length of 6 cm. Two sets of perpendicular axes, \(x-y\) and \(x'-y'\), are illustrated. The angle between the vector \(\overrightarrow{OP}\) and each axis is \(30^\circ\).The task is to express \(\overrightarrow{OP}\) in terms of its x and y components for each set of axes.(a) **Calculate the projections of \(\overrightarrow{OP}\) along the \(x\) and \(y\) directions.**- \((\overrightarrow{OP})_x =\) [input box] cm- \((\overrightarrow{OP})_y =\) [input box] cm(b) **Calculate the projections of \(\overrightarrow{OP}\) along the \(x'\) and \(y'\) directions.**- \((\overrightarrow{OP})_{x'} =\) [input box] cm- \((\overrightarrow{OP})_{y'} =\) [input box] cm---**Explanation of the Diagram:**The diagram features a vector labeled \(\overrightarrow{OP}\), originating from point \(O\) and extending towards point \(P\). The vector makes a \(30^\circ\) angle with each of the axes it crosses. The axes \(x-y\) and \(x'-y'\) are perpendicular to each other.The objective is to determine the vector's projections on both sets of axes. This involves calculating the horizontal and vertical components of the vector as aligned with each axis system.

Since you have posted a question with a multiple sub parts, we will solve first three sub - parts…

1. The vector OP shown in the figure has a length of 6 cm. Two sets of perpendicular axes, x-y and x'-y', are shown. Express OP in terms of its x and y components in each set of axes. (a) Calculate the projections of OP along the x and y directions. (OP)x= (OP)y= cm (OP) = (OP)y= cm 30° 30° X (b) Calculate the projections of OP along the x' and y' directions. cm ✔cm

1. The vector OP shown in the figure has a length of 6 cm. Two sets of perpendicular axes, x-y and x'-y', are shown. Express OP in terms of its x and y components in each set of axes. (a) Calculate the projections of OP along the x and y directions. (OP)x= (OP)y= cm (OP) = (OP)y= cm 30° 30° X (b) Calculate the projections of OP along the x' and y' directions. cm ✔cm

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Scalars and Vectors

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Question

### Vector Magnitude Calculation

#### (c) Use the projections of vector \(\overrightarrow{OP}\) along the x and y directions to calculate the magnitude of \(\overrightarrow{OP}\) using:
\[
OP = \sqrt{(OP_x)^2 + (OP_y)^2}
\]
\[
OP = \_\_\_\_ \text{ cm}
\]

#### (d) Use the projections of vector \(\overrightarrow{OP}\) along the x' and y' directions to calculate the magnitude of \(\overrightarrow{OP}\) using:
\[
OP = \sqrt{(OP_{x'})^2 + (OP_{y'})^2}
\]
\[
OP = \_\_\_\_ \text{ cm}
\]

#### (e) Conclusion
What can you conclude about the magnitude of a vector with respect to rotated coordinate axes?

**Conclusion:** The magnitude of the vectors using components in different coordinate axes that are rotated with respect to each other remains unchanged.

The image presents a vector problem involving two sets of perpendicular axes. Here's the transcription:

---

**1.**

The diagram displays a vector \(\overrightarrow{OP}\) with a length of 6 cm. Two sets of perpendicular axes, \(x-y\) and \(x'-y'\), are illustrated. The angle between the vector \(\overrightarrow{OP}\) and each axis is \(30^\circ\).

The task is to express \(\overrightarrow{OP}\) in terms of its x and y components for each set of axes.

(a) **Calculate the projections of \(\overrightarrow{OP}\) along the \(x\) and \(y\) directions.**

- \((\overrightarrow{OP})_x =\) [input box] cm
- \((\overrightarrow{OP})_y =\) [input box] cm

(b) **Calculate the projections of \(\overrightarrow{OP}\) along the \(x'\) and \(y'\) directions.**

- \((\overrightarrow{OP})_{x'} =\) [input box] cm
- \((\overrightarrow{OP})_{y'} =\) [input box] cm

---

**Explanation of the Diagram:**

The diagram features a vector labeled \(\overrightarrow{OP}\), originating from point \(O\) and extending towards point \(P\). The vector makes a \(30^\circ\) angle with each of the axes it crosses. The axes \(x-y\) and \(x'-y'\) are perpendicular to each other.

The objective is to determine the vector's projections on both sets of axes. This involves calculating the horizontal and vertical components of the vector as aligned with each axis system.

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