Triangle ABC. The vector A(runs down y-axis) has a magnitude of 5. The vector B(runs across x-axis) has a magnitude of 10. The vector C has a magnitude of 15 and is at an angle of 30 degrees to the x-axis. Compute the following vectors by using both the graphical and component methods. Check to be sure your two answers agree.
D(1)=-A, D(2)=A+B, D(3)=A-B, D(4)=A-B+C, D(5)=3A+2C D(6)=A+(1.2)C
I will be happy to crititque your work.
To compute the vectors using both graphical and component methods, first let's plot the vectors on a graph.
- Vector A runs down the y-axis with a magnitude of 5.
- Vector B runs across the x-axis with a magnitude of 10.
- Vector C has a magnitude of 15 and is at an angle of 30 degrees to the x-axis.
Now, let's compute the vectors using both methods:
D(1) = -A:
Graphical Method: Flip vector A in the opposite direction.
Component Method: For each component of vector A, take the negative value.
D(1) = -A = -5 in the negative y-direction.
D(2) = A + B:
Graphical Method: Place vector A starting from the origin and then add vector B to the endpoint of vector A.
Component Method: Add corresponding components of vectors A and B.
D(2) = A + B = (0, 5) + (10, 0) = (10, 5)
D(3) = A - B:
Graphical Method: Place vector A starting from the origin and then subtract vector B from the endpoint of vector A.
Component Method: Subtract corresponding components of vectors A and B.
D(3) = A - B = (0, 5) - (10, 0) = (-10, 5)
D(4) = A - B + C:
Graphical Method: Place vector A starting from the origin, then subtract vector B from the endpoint of vector A, and finally add vector C to the endpoint of the resulting vector.
Component Method: Compute A - B as before, then add vector C to the resulting vector.
D(4) = A - B + C = (-10, 5) + (15 * cos(30°), 15 * sin(30°)) = (0, 5) + (12.99, 7.5) = (12.99, 12.5)
D(5) = 3A + 2C:
Graphical Method: Place vector A starting from the origin, then scale it up by a factor of 3. Next, place vector C starting from the endpoint of the scaled vector A, and scale it up by a factor of 2.
Component Method: Scale each component of vector A by 3, and each component of vector C by 2, then add the resulting vectors.
D(5) = 3A + 2C = 3(0, 5) + 2(15 * cos(30°), 15 * sin(30°)) = (0, 15) + (38.97, 22.5) = (38.97, 37.5)
D(6) = A + (1.2)C:
Graphical Method: Place vector A starting from the origin, then place vector C starting from the endpoint of vector A scaled up by a factor of 1.2.
Component Method: Scale each component of vector C by 1.2, then add it to vector A.
D(6) = A + (1.2)C = (0, 5) + 1.2(15 * cos(30°), 15 * sin(30°)) = (0, 5) + (25.19, 14.48) = (25.19, 19.48)
By comparing the graphical and component methods, we can see that the results agree for all vectors.