In the sum Vector A plus Vector B equals vector C, vector A has a magnitude of 11.6 m and is angled 36.5° counterclockwise from the +x direction, and vector C has a magnitude of 14.0 m and is angled 17.1° counterclockwise from the -x direction. What are (a) the magnitude and (b) the angle (relative to +x) of Vector B.
To find the magnitude and angle of Vector B, we can use vector addition and trigonometry. The formula for vector addition is:
C = A + B
where A and B are given vectors, and C is the resultant vector.
(a) To find the magnitude of Vector B, we need to rearrange the formula as:
B = C - A
Substituting the given values, we have:
|C| = 14.0 m (magnitude of C)
|A| = 11.6 m (magnitude of A)
So, the magnitude of Vector B is:
|B| = |C| - |A| = 14.0 m - 11.6 m = 2.4 m
Therefore, the magnitude of Vector B is 2.4 m.
(b) To find the angle of Vector B, we can use trigonometry. We can break down Vector B into its x-component and y-component. The x-component can be found by subtracting the x-component of Vector A from the x-component of Vector C, and the y-component can be found by subtracting the y-component of Vector A from the y-component of Vector C.
Let's start by finding the x-component of Vector B:
Bx = Cx - Ax
where Cx and Ax are the x-components of vectors C and A, respectively. Since Vector C is angled 17.1° counterclockwise from the -x direction, and Vector A is angled 36.5° counterclockwise from the +x direction, we need to find the x-components using trigonometric functions:
Cx = |C| * cos(angle of C)
Ax = |A| * cos(angle of A)
Substituting the given values:
Cx = 14.0 m * cos(17.1°)
Ax = 11.6 m * cos(36.5°)
Calculating these values, we get:
Cx ≈ 13.481 m
Ax ≈ 9.517 m
Now, we can calculate the x-component of Vector B:
Bx = Cx - Ax ≈ 13.481 m - 9.517 m ≈ 3.964 m
Next, let's find the y-component of Vector B:
By = Cy - Ay
using the y-components of vectors C and A:
Cy = |C| * sin(angle of C)
Ay = |A| * sin(angle of A)
Substituting the given values:
Cy = 14.0 m * sin(17.1°)
Ay = 11.6 m * sin(36.5°)
Calculating these values, we get:
Cy ≈ 4.860 m
Ay ≈ 6.785 m
Now, we can calculate the y-component of Vector B:
By = Cy - Ay ≈ 4.860 m - 6.785 m ≈ -1.925 m
Finally, we can find the angle (θ) of Vector B relative to the +x direction using the arctan function:
θ = arctan(By / Bx)
Substituting the calculated values:
θ = arctan(-1.925 m / 3.964 m)
Calculating this value, we get:
θ ≈ -25.674°
Note that the negative sign indicates that Vector B is angled clockwise from the +x direction.
Therefore, the magnitude of Vector B is 2.4 m, and the angle of Vector B relative to the +x direction is approximately -25.674° (clockwise).