Finding Magnitudes of Displacement Vectors in Physics Homework

[PhilCam]

This is my first semester of physics as I did not take it in high school and I am having a difficult time with this homework question. Any help would be greatly appreciated.

Homework Statement

The route followed by a hiker consists of three displacement vectors A, B, and C. Vector A is along a measured trail and is 1550 m in a direction 24.0° north of east. Vector B is not along a measured trail, but the hiker uses a compass and knows that the direction is 41.0° east of south. Similarly, the direction of vector C is 18.0° north of west. The hiker ends up back where she started, so the resultant displacement is zero, or vector A + vector B + vector C = 0. Find the magnitudes of vector B and vector C.

Homework Equations

Vector A = Ax (x) + Ay (y)

The Attempt at a Solution

I tried to solve this problem several different ways. Firstly I used the angle and m to form a right angle and used sin and cos to find Ax and Ay. Ax=630.44m Ay=1416m. After I figured this out I was unsure how to continue with finding the solution because I did not know how to find Bx, By, or Cx, and Cy.

Next, I tried to draw a triangle coming back to the (0,0) by connecting the vectors. However, when I drew this out, I somehow ended up with angles of 41, 24, and 18 which are obviously way too small to make the 180 degrees necessary for a triangle.

Once again, any help would be greatly appreciated. Thanks.

 

[berkeman]

This is my first semester of physics as I did not take it in high school and I am having a difficult time with this homework question. Any help would be greatly appreciated.

Homework Statement

The route followed by a hiker consists of three displacement vectors A, B, and C. Vector A is along a measured trail and is 1550 m in a direction 24.0° north of east. Vector B is not along a measured trail, but the hiker uses a compass and knows that the direction is 41.0° east of south. Similarly, the direction of vector C is 18.0° north of west. The hiker ends up back where she started, so the resultant displacement is zero, or vector A + vector B + vector C = 0. Find the magnitudes of vector B and vector C.

Homework Equations

Vector A = Ax (x) + Ay (y)

The Attempt at a Solution

I tried to solve this problem several different ways. Firstly I used the angle and m to form a right angle and used sin and cos to find Ax and Ay. Ax=630.44m Ay=1416m. After I figured this out I was unsure how to continue with finding the solution because I did not know how to find Bx, By, or Cx, and Cy.

Next, I tried to draw a triangle coming back to the (0,0) by connecting the vectors. However, when I drew this out, I somehow ended up with angles of 41, 24, and 18 which are obviously way too small to make the 180 degrees necessary for a triangle.

Once again, any help would be greatly appreciated. Thanks.

Draw vector A on graph paper. Then draw a line from the end of A in the direction of the B vector. Then draw a line from the origin, backwards along the direction of the C vector. That forms the triangle, and you should be able to figure out the rectangular displacements in X and Y that each vector forms. Does that help?

Welcome to the PF, BTW.

 

[PhilCam]

I'm not quite sure I follow. If I draw the vectors together, the first angle would be the initial 24 degrees in the northeast direction then the second angle would be the 41 degrees in the southeast direction and the final angle of the triangle would be 115?

180 = (24+41+X)
180 = 65 +X
X=115

 

[berkeman]

I'm not quite sure I follow. If I draw the vectors together, the first angle would be the initial 24 degrees in the northeast direction then the second angle would be the 41 degrees in the southeast direction and the final angle of the triangle would be 115?

180 = (24+41+X)
180 = 65 +X
X=115

Not exactly. The first vector is given to you as 24 degrees North of East. That's not 24 degrees Northeast, necessarily. You have to be careful to draw it correcdtly. They also give you the magnitude of that displacement, so that fixes the end point.

The second is 41 degrees South of East, so be sure to draw that direction correctly. You don't know yet how long B is, so just draw a long line.

The third line will hit the origin, and is aimed from a point along the B line somewhere, so that the angle it makes with the horizontal is 18 degrees (North of West). Those directions and that first point at the end of A define a triangle. Do you see that now?

 

[berkeman]

EDIT -- deleted my incorrect comment about what the interior angles of a triangle add up to. 180 is the correct number. Sorry for any confusion!

 

[PhilCam]

I have no idea how to find the internal angles. I understand the first one will be 24 degrees, but after that I'm totally lost. I've been trying to figure it out for the last 45 minutes.

 

[berkeman]

I have no idea how to find the internal angles. I understand the first one will be 24 degrees, but after that I'm totally lost. I've been trying to figure it out for the last 45 minutes.

It's easiest to visualize with a drawing. Draw an x-y axis (North at top, etc), and draw the lines using the directions given with respect to the compass directions...

 

[PhilCam]

It's easiest to visualize with a drawing. Draw an x-y axis (North at top, etc), and draw the lines using the directions given with respect to the compass directions...

Right I did draw it and I ended up with angles of 24, 115, and 41. When I used this angles to find the magnitudes, I got the wrong answer.

I have drew them together on an x-y axis and this is what I have ended up with.

 

[berkeman]

Right I did draw it and I ended up with angles of 24, 115, and 41. When I used this angles to find the magnitudes, I got the wrong answer.

I have drew them together on an x-y axis and this is what I have ended up with.

How can the first interior angle be 24 degrees, when the bottom right of the triangle returns to the origin from below the initial horizontal? That angle has to be 24 plus something degrees. Probably related to the final direction, which is given to you as 18 degrees North of West...