A hiker follows a route that consists of three displacement vectors A, B, and C. Vector A is along a measured trail and is 1550 meters in a direction 25 degrees north of east. The direction of Vector B is 41 degrees east of south, and the direction of vector C is 35 degrees north of west. The hiker ends up back where at the original starting point, so the reslultant displacement is zero or A+B+C=0. Find the magnitudes of Vectors B and C.

Question

A hiker follows a route that consists of three displacement vectors A, B, and C. Vector A is along a measured trail and is 1550 meters in a direction 25 degrees north of east. The direction of Vector B is 41 degrees east of south, and the direction of vector C is 35 degrees north of west. The hiker ends up back where at the original starting point, so the reslultant displacement is zero or A+B+C=0. Find the magnitudes of Vectors B and C.

An explanation of the answer with some work would be greatly appreciated. Thanks.

Answer 1

Break each of the vectors into N, E components.

For instance:
vectorC is -C*Cos35 E and C*sin35N
do that for all three vectors, then the sum of all three E vectors will be zero, and the sum of all three N vectors will be zero. Solve B, C (B,C is a magnitude of vectors b, C

Answer 2

To solve this problem, we can use vector addition. Since the resultant displacement is zero (A + B + C = 0), we can rearrange the equation to solve for vector A.

A = - (B + C)

Now let's calculate vector B.

The magnitude of vector B can be found using the formula:

|B| = √(Bx² + By²)

where Bx is the horizontal component and By is the vertical component of vector B. To find these values, we need to use the given information about the direction of vector B.

Direction B: 41 degrees east of south.

First, let's determine the angle measured from the eastward direction. Remember that the eastward direction is 0 degrees.

Angle from the eastward direction = 90 degrees - 41 degrees = 49 degrees

Now, we can find Bx and By using trigonometry:

Bx = |B| × cos(angle from the eastward direction)
By = |B| × sin(angle from the eastward direction)

Next, let's calculate vector C.

The magnitude of vector C can be found using the same formula:

|C| = √(Cx² + Cy²)

where Cx is the horizontal component and Cy is the vertical component of vector C. We need to use the given information about the direction of vector C.

Direction C: 35 degrees north of west.

First, let's determine the angle measured from the westward direction. Remember the westward direction is 180 degrees.

Angle from the westward direction = 180 degrees - 35 degrees = 145 degrees

Now, we can find Cx and Cy using trigonometry:

Cx = |C| × cos(angle from the westward direction)
Cy = |C| × sin(angle from the westward direction)

Finally, we can substitute the values of B and C into the equation A = -(B + C) and solve for vector A.

A = - (B + C)
A = - (Bx î + By ĵ) - (Cx î + Cy ĵ)
A = - ((Bx + Cx) î + (By + Cy) ĵ)

Since the resultant displacement A is zero, we know that both the horizontal (î) and vertical (ĵ) components of vector A must be equal to zero.

Therefore, we can set:

Bx + Cx = 0
By + Cy = 0

Now we have two equations with two unknowns (Bx, By) and (Cx, Cy). We can solve these equations to find the values of Bx, By, Cx, and Cy.

Once we have the values of Bx and Cx, we can substitute them back into the formulas for B and C to find their magnitudes |B| and |C|.

I hope this explanation helps you understand how to approach and solve the problem step by step.