Three vectors a, b, and c, each have a magnitude of 42 m and lie in an xy plane. Their directions relative to the positive direction of the x axis are 27 ˚, 194 ˚, and 312 ˚, respectively. B) i need the angle of the vector a+b+c (relative to the +x direction in the range of (-180°, 180°) i got 42 but its not correct do i have to add it to 180? D)angle of a-b+c in the range of (-180°, 180°) i got 5 degrees for this. and it is incorrect
a+b+c = (42cos27,42sin27) + (42cos194, 42sin194) + (42cos312,42sin312)
= (24.7733, -22.3052)
magnitude = √(24.7733^2 + 22.3052^2) = appr 33.335
direction angle Ø, such that
tanØ = -22.3052/24.7733
I get Ø = appr -41.998°
or 318.00°
compare with your work, and find your error.
how is -41.998 = to 318?
Displacement = 42m[27] + 42[194] + 42[312].
X = 42*Cos27 + 42*Cos194 + 42*Cos312 = 24.77 m.
Y = 42*sin27 + 42*sin194 + 42*sin312 = -22.3 m.
X + Yi = 24.77 - 22.3i = 33.33m[-42o] = 33.33m[42o] S. of E. = 33.33m[318o]CCW.
Angle calculations:
Tan A = (-22.30/24.77 = -0.90028,
A = -42o, Q4.
360-42 = 318o CCW from +x-axis.
so, if the x-component is positive and the y-component is negative it will end up in the fourth quadrant? therefore, I would always have to subtract it by 360?
nvm i got it thanks
To find the angle of vector a + b + c relative to the positive x-direction, you need to add up the individual angles of a, b, and c. Here's how you can calculate it step by step:
Step 1: Calculate the x and y components of each vector.
- For vector a, the x-component can be found using the formula: ax = magnitude * cos(angle). Therefore, ax = 42 m * cos(27°).
Similarly, the y-component of a is ay = 42 m * sin(27°).
- Repeat the same process for vectors b and c to find their x and y components.
Step 2: Add up the x and y components of a, b, and c.
- Add the x-components of a, b, and c to get the total x-component (let's call it rx).
- Similarly, add the y-components of a, b, and c to get the total y-component (let's call it ry).
Step 3: Calculate the magnitude of the resultant vector.
- The magnitude of the resultant vector R can be found using the formula: R = sqrt(rx^2 + ry^2).
Step 4: Calculate the angle of the resultant vector.
- The angle of the resultant vector can be found using the formula: angle = atan2(ry, rx).
Now, let's apply this process to find the angle of vector a + b + c:
Step 1: Calculate the components of vectors a, b, and c:
- For vector a: ax = 42 m * cos(27°) ≈ 37.97 m and ay = 42 m * sin(27°) ≈ 18.37 m.
- For vector b: bx = 42 m * cos(194°) ≈ -10.15 m and by = 42 m * sin(194°) ≈ -40.95 m.
- For vector c: cx = 42 m * cos(312°) ≈ -34.07 m and cy = 42 m * sin(312°) ≈ -23.50 m.
Step 2: Add up the x and y components:
- Sum of x-components: rx = ax + bx + cx = 37.97 m + (-10.15 m) + (-34.07 m) ≈ -6.25 m
- Sum of y-components: ry = ay + by + cy = 18.37 m + (-40.95 m) + (-23.50 m) ≈ -46.08 m
Step 3: Calculate the magnitude of the resultant vector:
- R = sqrt(rx^2 + ry^2) = sqrt((-6.25 m)^2 + (-46.08 m)^2) ≈ 46.51 m
Step 4: Calculate the angle of the resultant vector:
- angle = atan2(ry, rx) ≈ atan2(-46.08 m, -6.25 m) ≈ -83.92°
Therefore, the angle of vector a + b + c relative to the positive x-direction is approximately -83.92°.
Now let's move on to calculating the angle of vector a - b + c:
Step 1: Calculate the components of vectors a, b, and c (same as before).
Step 2: Add up the x and y components of a, b, and c.
Step 3: Calculate the magnitude of the resultant vector (same formula as before).
Step 4: Calculate the angle of the resultant vector (same as before).
Using the same steps, you can find the angle of vector a - b + c.