Graphic Artist (Vector Problem)

[Hi.O303]

Homework Statement

The origin ( 0 , 0 ) is at the upper left corner of the image, the +x-axis points to the right, the +y-axis point down. Distances are measured in pixels. The artist draws a line from the pixel location ( 10 , 20 ) to the location ( 210, 200 ). She wishes to draw a second line that starts at ( 10 , 20 ), which is 250 pixels long, and is at angle 30 degrees clockwise from the first line.
(a) At which pixel location should this second line end? Give your answer to the nearest pixel.

Homework Equations

sinx = opp/hyp , cosx = adj/hyp , tan = opp/adj
a^2+b^2 = c^2

The Attempt at a Solution

First to get the angle of the first line : I subtract 210 - 10 = 200 = x , 200 - 20 = 180 = y .
Then to get the angle of the first line I take arctan ( 180 / 200 ) = 41.9872° , So the angle of the second line must be 71.9468° . Then since the magnitude of the vector is 250, to get the component vectors I just take
250sin(71.9872) = y = 237.7468 + 20 = 257.7468
250cos(71.9872) = x = 87.3073 + 10 = 87.3073
So I guess they want the area that is covered, which is
(87.3073² + 257.7468²) = 64146 pixels , but their answer is 87,258. Does anybody know how they could've gotten this answer?

 

[anathema]

Hello,

To find the ending pixel location of the second line, we can use the trigonometric functions sine and cosine to calculate the x and y components of the vector.

First, we know that the x component will be 250*cos(30°) = 250*(√3/2) = 216.5064 pixels.

Next, we can use the Pythagorean theorem to find the y component. Since we know the magnitude of the vector is 250 pixels, and the x component is 216.5064 pixels, we can solve for the y component:

250² = 216.5064² + y²
y² = 250² - 216.5064²
y = √(250² - 216.5064²) = 87.3073 pixels

Thus, the ending pixel location for the second line is (10 + 216.5064, 20 + 87.3073) = (226.5064, 107.3073).

I hope this helps clarify the solution for you!

 

[tomcue]

I would approach this problem by first clarifying the specific goal of the second line. Is it meant to be parallel to the first line, or at a specific angle relative to the first line? This information would impact the calculation of the endpoint of the second line.

Assuming that the second line is meant to be at a specific angle relative to the first line, the calculation provided in the attempt at a solution is correct. However, there may be a rounding error in the calculation, which could explain the discrepancy between the calculated endpoint and the given answer.

To double-check the calculations, I would recommend using a calculator or computer program that can handle more decimal places to avoid rounding errors. Additionally, it may be helpful to draw a diagram to visualize the problem and confirm the calculated endpoint.

If the discrepancy cannot be resolved, it may be worth reaching out to the author of the problem for clarification or to confirm the given answer. As a scientist, it is important to ensure accuracy and precision in our calculations and results.