Vector V1 is 6.6 units long and points along the negative x axis. V2 is 8.5 units long and points at +55 degrees to the positive x axis.
A) What are the x and y components of each vector?
B) Determine the sum of vector 1 and vector 2. (magnitude and angle)
I have solved the components for V2, but I am stuck on solving for the components of V1 because I can't figure out how to solve for the angle of V1.
I don't now
huh? v1 = <-6.6,0>
don't get so bogged won in formulas that you fail to recognize simple geometry!
If you must rely on the formulas, note that θ=180°
To solve for the components of V1, we can use the fact that it points along the negative x-axis. Since it is a straight line pointing directly left, the angle it makes with the positive x-axis is 180 degrees.
A) Components of V1:
Since V1 points along the negative x-axis, the x-component will be negative and the y-component will be zero.
x-component of V1: -6.6 units
y-component of V1: 0 units
Now, let's move on to calculating the components of V2.
Since we know that the magnitude of V2 is 8.5 units and it makes an angle of +55 degrees with the positive x-axis, we can use trigonometry to find its x and y components.
The angle between the vector and the x-axis is given as +55 degrees, which means we need to measure it counterclockwise from the positive x-axis.
A) Components of V2:
To find the x-component of V2:
x-component of V2 = magnitude of V2 * cos(angle)
x-component of V2 = 8.5 units * cos(55 degrees)
x-component of V2 ≈ 8.5 * 0.5736
x-component of V2 ≈ 4.86 units
To find the y-component of V2:
y-component of V2 = magnitude of V2 * sin(angle)
y-component of V2 = 8.5 units * sin(55 degrees)
y-component of V2 ≈ 8.5 * 0.8192
y-component of V2 ≈ 6.96 units
So, the components of V2 are approximately:
x-component of V2 ≈ 4.86 units
y-component of V2 ≈ 6.96 units
B) Sum of V1 and V2:
To find the sum of V1 and V2, we can add their corresponding components together.
x-component of the sum = x-component of V1 + x-component of V2 = -6.6 units + 4.86 units = -1.74 units
y-component of the sum = y-component of V1 + y-component of V2 = 0 units + 6.96 units = 6.96 units
To find the magnitude:
magnitude of the sum = sqrt((x-component of the sum)^2 + (y-component of the sum)^2)
magnitude of the sum = sqrt((-1.74 units)^2 + (6.96 units)^2)
magnitude of the sum = sqrt(3.0276 units^2 + 48.4416 units^2)
magnitude of the sum ≈ sqrt(51.4692 units^2)
magnitude of the sum ≈ 7.18 units
To find the angle:
angle of the sum = arctan(y-component of the sum / x-component of the sum)
angle of the sum = arctan(6.96 units / -1.74 units)
angle of the sum = arctan(-3.999)
angle of the sum ≈ -75.97 degrees
Therefore, the sum of V1 and V2 is approximately:
magnitude of the sum ≈ 7.18 units
angle of the sum ≈ -75.97 degrees
To solve for the components of V1, you need to determine the angle it makes with the positive x-axis. Since V1 points along the negative x-axis, it forms an angle of 180 degrees or π radians with the positive x-axis.
A) Components of V1:
Since V1 points along the negative x-axis, its x-component will be negative and its y-component will be 0.
x-component of V1 = -6.6
y-component of V1 = 0
Now, let's move on to solving for B.
B) Sum of V1 and V2:
To find the sum of the two vectors, you need to add their x-components and y-components separately.
x-component of V1 + V2 = -6.6 + (8.5 * cos(55°))
y-component of V1 + V2 = 0 + (8.5 * sin(55°))
To calculate the values, you can use the formulas:
cos(θ) = adjacent/hypotenuse
sin(θ) = opposite/hypotenuse
Using the given information, we can substitute the values:
x-component of V1 + V2 = -6.6 + (8.5 * cos(55°))
y-component of V1 + V2 = 0 + (8.5 * sin(55°))
Once you calculate the values for the x and y components, you can determine the magnitude and angle of the resultant vector using the Pythagorean theorem:
Magnitude of the resultant vector = sqrt((x-component of V1 + V2)^2 + (y-component of V1 + V2)^2)
Angle of the resultant vector = arctan((y-component of V1 + V2)/(x-component of V1 + V2))
I hope this helps! Let me know if you have any further questions.