Welcome to Westonci.ca, your go-to destination for finding answers to all your questions. Join our expert community today! Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To solve for the direction of the deer's resultant vector, follow these steps:
1. Vector Components Breakdown:
- The deer initially runs 88 feet directly to the east.
- Then, the deer turns and runs \( 127 \) feet at an angle of \( 11^{\circ} \) north of west.
2. Coordinate System Setup:
- Consider the east direction as the positive x-axis and the north direction as the positive y-axis.
- Decompose the \( 127 \)-foot run into its northward (y) and westward (x) components.
3. Calculating Westward and Northward Components:
- To find the westward distance (x-component):
[tex]\[ x = 127 \cos(11^{\circ}) \][/tex]
- To find the northward distance (y-component):
[tex]\[ y = 127 \sin(11^{\circ}) \][/tex]
4. Resultant Components:
- The total x-component (westward minus eastward distance):
[tex]\[ \text{Resultant}_x = -88 + x = -88 + 127 \cos(11^{\circ}) \][/tex]
Note that east is positive and west is negative in this coordinate system.
- The northward distance remains the same as the y-component found:
[tex]\[ \text{Resultant}_y = y = 127 \sin(11^{\circ}) \][/tex]
5. Magnitude of the Resultant Vector:
- Using the Pythagorean theorem to calculate the magnitude:
[tex]\[ |\vec{R}| = \sqrt{\text{Resultant}_x^2 + \text{Resultant}_y^2} \][/tex]
6. Direction of the Resultant Vector:
- The direction angle, \( \theta \), north of west, is determined from the arctangent of the ratio of the northward and westward components:
[tex]\[ \theta = \tan^{-1}\left(\frac{\text{Resultant}_y}{|\text{Resultant}_x|}\right) \][/tex]
Note that \( \text{Resultant}_x \) is negative (west), so the angle calculated will be in the correct north of west orientation.
Given all the above steps and values:
- The magnitude of the resultant vector \( |\vec{R}| \) is approximately \( 43.95 \) feet.
- The direction \(\theta\) is approximately \( 33.46^{\circ} \) north of west.
Hence, the resultant direction \( \theta \) of the deer's resultant vector is approximately:
[tex]\[ \boxed{33.46^{\circ}} \][/tex]
1. Vector Components Breakdown:
- The deer initially runs 88 feet directly to the east.
- Then, the deer turns and runs \( 127 \) feet at an angle of \( 11^{\circ} \) north of west.
2. Coordinate System Setup:
- Consider the east direction as the positive x-axis and the north direction as the positive y-axis.
- Decompose the \( 127 \)-foot run into its northward (y) and westward (x) components.
3. Calculating Westward and Northward Components:
- To find the westward distance (x-component):
[tex]\[ x = 127 \cos(11^{\circ}) \][/tex]
- To find the northward distance (y-component):
[tex]\[ y = 127 \sin(11^{\circ}) \][/tex]
4. Resultant Components:
- The total x-component (westward minus eastward distance):
[tex]\[ \text{Resultant}_x = -88 + x = -88 + 127 \cos(11^{\circ}) \][/tex]
Note that east is positive and west is negative in this coordinate system.
- The northward distance remains the same as the y-component found:
[tex]\[ \text{Resultant}_y = y = 127 \sin(11^{\circ}) \][/tex]
5. Magnitude of the Resultant Vector:
- Using the Pythagorean theorem to calculate the magnitude:
[tex]\[ |\vec{R}| = \sqrt{\text{Resultant}_x^2 + \text{Resultant}_y^2} \][/tex]
6. Direction of the Resultant Vector:
- The direction angle, \( \theta \), north of west, is determined from the arctangent of the ratio of the northward and westward components:
[tex]\[ \theta = \tan^{-1}\left(\frac{\text{Resultant}_y}{|\text{Resultant}_x|}\right) \][/tex]
Note that \( \text{Resultant}_x \) is negative (west), so the angle calculated will be in the correct north of west orientation.
Given all the above steps and values:
- The magnitude of the resultant vector \( |\vec{R}| \) is approximately \( 43.95 \) feet.
- The direction \(\theta\) is approximately \( 33.46^{\circ} \) north of west.
Hence, the resultant direction \( \theta \) of the deer's resultant vector is approximately:
[tex]\[ \boxed{33.46^{\circ}} \][/tex]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.