For the pair of vectors shown on the right; find components of the resultant vector along with the magnitude and direction angle of the resultant R=v+ “-D What is the magnitude? IRI (Round to the nearest hundredth ) What is the direction angle? CR (Round to the nearest tenth as needed )
The Correct Answer and Explanation is:
Here are the final answers:
v = 3 i + 6 j
u = 7 i + 3 j
R = v + u = 10 i + 9 j
What is the magnitude? |R| = 13.45
(Round to the nearest hundredth.)
What is the direction angle? θ_R = 42.0 °
(Round to the nearest tenth as needed.)
Explanation
This problem requires us to determine the properties of a resultant vector, which is the sum of two other vectors shown on a graph. The process involves three main steps: identifying the components of the initial vectors, calculating the components of their sum, and then finding the magnitude and direction of that sum.
1. Finding the Components of v and u
First, we determine the components of vectors v (the red vector) and u (the blue vector) from the provided graph. A vector in the form xi + yj has a horizontal component x and a vertical component y. By observing the graph, we can see that both vectors start at the origin (0, 0).
- For vector v, its tip is at the coordinate (3, 6). This means it extends 3 units along the x-axis and 6 units along the y-axis. Therefore, v = 3i + 6j.
- For vector u, its tip is at the coordinate (7, 3). This means it extends 7 units along the x-axis and 3 units along the y-axis. Therefore, u = 7i + 3j.
2. Calculating the Resultant Vector R
The resultant vector R is the sum of v and u. To add vectors, we simply add their corresponding components.
- R = v + u = (3i + 6j) + (7i + 3j)
- R = (3 + 7)i + (6 + 3)j
- R = 10i + 9j
3. Calculating the Magnitude and Direction of R
Now that we have the components of R (Rx = 10, Ry = 9), we can find its magnitude and direction angle.
- Magnitude: The magnitude |R| is calculated using the Pythagorean theorem: |R| = √(Rx² + Ry²).
- |R| = √(10² + 9²) = √(100 + 81) = √181
- |R| ≈ 13.4536. Rounded to the nearest hundredth, the magnitude is 13.45.
- Direction Angle: The direction angle θ_R is found using the arctangent function: θ_R = arctan(Ry / Rx).
- θ_R = arctan(9 / 10) = arctan(0.9)
- θ_R ≈ 41.987°. Since both components are positive, the vector is in the first quadrant. Rounded to the nearest tenth, the direction angle is 42.0°.
