Can someone show the steps involved in figuring out this type of problem?
Given the magnitudes of vectors u and v and the angle Ό between them, find the magnitude of the sum u + v to the
nearest tenth and the angle that the sum vector makes with u to the nearest degree.
/u/= 20, /v/= 86, Ό = 90°
The answer is 88.3 and 77deg but I don't know how to get to this. Thanks in advance
Follow the same method and steps as I showed you in
http://www.jiskha.com/display.cgi?id=1311551145
except this one is easier since it forms a right-angled triangle, and your parallogram is a rectangle.
If the resultant is R, then
R^2 = 20^2 + 86^2
= 7796
R = √7796 = 88.29
Now use the Sine Law again to get the angle between vectors
X = 20.
Y = 86.
tanA = Y/X = 86/20 = 4.3,
A = 76.9 deg.
R = X / cosA = 20 / cos76.9 = 88.24
OR
R = Y / sinA = 86 / sin76.9 = 88.3
.
To solve this problem, you can use the concept of vector addition and trigonometry. Here are the steps involved:
1. Draw a diagram: A diagram helps visualize the given vectors u and v and their angles. Draw two vectors u and v originating from the same point.
2. Find the x and y components of vectors u and v: Since u and v are given in terms of their magnitudes (/u/ and /v/) and the angle Ό, you can find their horizontal and vertical components using trigonometry. Let's consider u first:
- The horizontal component of u (u_x) can be found using the equation: u_x = /u/ * cos(Ό).
- The vertical component of u (u_y) can be found using the equation: u_y = /u/ * sin(Ό).
Similarly, you can find the x and y components of vector v (v_x and v_y) using the same formulas.
3. Add the x and y components of vectors u and v: To find the x and y components of the vector sum (u + v), add the corresponding components of vectors u and v. This can be done using simple arithmetic. Let's call the x and y components of the vector sum (u + v) as R_x and R_y:
- R_x = u_x + v_x
- R_y = u_y + v_y
4. Find the magnitude of the sum vector (u + v): Using the x and y components of the sum vector (R_x and R_y), you can find its magnitude using the Pythagorean theorem:
- The magnitude of the sum vector, denoted as |u + v|, can be found using the equation: |u + v| = sqrt(R_x^2 + R_y^2).
5. Find the angle that the sum vector makes with u: To find the angle between the sum vector (u + v) and vector u, you can use the inverse tangent function (arctan). Let's call this angle α:
- α = arctan(R_y / R_x)
6. Calculate the final answers: Substituting the values of R_x and R_y into the equations from step 4 and step 5, you can calculate the magnitude of the sum vector (|u + v|) and the angle it makes with vector u (α). Round these values to the nearest tenth and degree, respectively.
By following these steps, you should arrive at the correct answers of 88.3 and 77 degrees for the magnitude and angle, respectively.