The route followed by a hiker consists of three displacement vectors , , and . Vector is along a measured trail and is 1060 m in a direction 20.0 ° north of east. Vector is not along a measured trail, but the hiker uses a compass and knows that the direction is 35.0 ° east of south. Similarly, the direction of vector is 17.0 ° north of west. The hiker ends up back where she started, so the resultant displacement is zero, or + + = 0. Find the magnitudes of (a) vector B and (b) vector C

(a) Ah, the magnitudes of vector B. Let's see what we have here. We know that vector B is not along a measured trail, which means it's probably enjoying its own little adventure. It's like that one friend who always goes off the beaten path, right?

Anyway, the hiker used a compass to determine the direction of vector B, which is 35.0 ° east of south. So, vector B took a bit of a detour, but that's okay. We'll figure out its magnitude.

Now, I can't just snap my fingers and magically calculate it for you, but I can give you a hint. Since the resultant displacement is zero, the magnitudes of the three vectors must add up to zero too. So, you can use that information to solve for the magnitude of vector B. Happy math-ing!

(b) And now, onto vector C. Vector C is feeling a bit rebellious, being 17.0 ° north of west. It's like it wants to break free from the norm, you know? Anyway, the hiker is back where she started, which means the resultant displacement is zero. So, we'll use the same princple as before.

Since the magnitudes of the three vectors must add up to zero, you can use that to find the magnitude of vector C. I believe in you!

To find the magnitudes of vector B and vector C, we can use the fact that the resultant displacement is zero. This means that the sum of the displacement vectors must be equal to zero:

A + B + C = 0

Let's first break down each vector into its components.

Vector A is given as 1060 m in a direction 20.0° north of east. To find its components, we can use trigonometry. The magnitude of A can be found using the Pythagorean theorem:

|A| = √(A_x^2 + A_y^2)

For vector A, the angle given is north of east, so we can split it into two components: A_x in the east direction and A_y in the north direction.

A_x = |A| * cos(20°)
A_y = |A| * sin(20°)

Using the given values, we can calculate the components of vector A:

A_x = 1060 * cos(20°)
A_y = 1060 * sin(20°)

Next, let's consider vector B. It is not along a measured trail, but we know its direction is 35.0° east of south. We can also split vector B into two components, one in the east direction and the other in the south direction. Let's call these components B_x and B_y.

B_x = |B| * cos(35°)
B_y = -|B| * sin(35°)

The negative sign for B_y is because the direction is south, which is opposite to the positive direction of the y-axis.

Finally, let's consider vector C. It has a direction of 17.0° north of west. We can similarly split it into two components, C_x in the west direction and C_y in the north direction.

C_x = -|C| * cos(17°)
C_y = |C| * sin(17°)

Now we can substitute these component values into the equation A + B + C = 0:

A_x + B_x + C_x = 0
A_y + B_y + C_y = 0

Plug in the component values we calculated earlier and solve the equations to find the magnitudes of vector B and vector C.