For the vectors a = (3.9 m)i + (4.8 m)j, and b = (5.4 m)i - (2.0 m)j, give the x and y components, magnitudes, and directions of the following vector operations.
(a.) a + b
______(x component)
______(y component)
______(magnitude)
______° (direction from the positive x axis)
(b.) b - a
______(x component)
______(y component)
______(magnitude)
______° (direction from the positive x axis)
I will be happy to check your answers, or critique your thinking. We don't do tests for students.
it's not a test. just a hw assignment one of a bunch of questions im stuck on.
~lor
No problem, I'm here to help! Let's break down the problem step by step.
First, let's find the x and y components of vector a and b:
For vector a:
x component = 3.9 m
y component = 4.8 m
For vector b:
x component = 5.4 m
y component = -2.0 m
Now, let's calculate the vector operations:
(a.) a + b:
To add two vectors, simply add their corresponding x and y components:
x component = (3.9 m) + (5.4 m) = 9.3 m
y component = (4.8 m) + (-2.0 m) = 2.8 m
Next, let's find the magnitude of the vector. The magnitude of a vector can be found using the Pythagorean theorem:
magnitude = sqrt((x component)^2 + (y component)^2)
magnitude = sqrt((9.3 m)^2 + (2.8 m)^2) ≈ 9.81 m
To find the direction of the vector from the positive x-axis, we can use trigonometry. The direction can be found using the inverse tangent function:
direction = atan2(y component, x component)
direction = atan2(2.8 m, 9.3 m) ≈ 17.47°
Therefore, for (a.) a + b:
x component = 9.3 m
y component = 2.8 m
magnitude ≈ 9.81 m
direction ≈ 17.47°
(b.) b - a:
To subtract two vectors, simply subtract their corresponding x and y components:
x component = (5.4 m) - (3.9 m) = 1.5 m
y component = (-2.0 m) - (4.8 m) = -6.8 m
Now, let's find the magnitude of the vector:
magnitude = sqrt((x component)^2 + (y component)^2)
magnitude = sqrt((1.5 m)^2 + (-6.8 m)^2) ≈ 7.04 m
To find the direction of the vector from the positive x-axis:
direction = atan2(y component, x component)
direction = atan2(-6.8 m, 1.5 m) ≈ -77.47°
Therefore, for (b.) b - a:
x component = 1.5 m
y component = -6.8 m
magnitude ≈ 7.04 m
direction ≈ -77.47°
I hope this helps! Let me know if you have any further questions.