example 1 using components to add vectors\nthree players on a reality tv show are brought to the center of a…

example 1 using components to add vectors\nthree players on a reality tv show are brought to the center of a large, flat field. each is given a meter stick, a compass, a calculator, a shovel, and (in a different order for each contestant) the following three displacements:\n$vec{a}$: 72 m, 32.0° east of north\n$vec{b}$: 57.3 m, 36.0° south of west\n$vec{c}$: 17.8 m due south\nthe three displacements lead to the point in the field where the keys to a new porsche are buried. two players start measuring immediately, but the winner first calculates where to go. what does she calculate?

example 1 using components to add vectors\nthree players on a reality tv show are brought to the center of a large, flat field. each is given a meter stick, a compass, a calculator, a shovel, and (in a different order for each contestant) the following three displacements:\n$vec{a}$: 72 m, 32.0° east of north\n$vec{b}$: 57.3 m, 36.0° south of west\n$vec{c}$: 17.8 m due south\nthe three displacements lead to the point in the field where the keys to a new porsche are buried. two players start measuring immediately, but the winner first calculates where to go. what does she calculate?

Answer

Explanation:

Step1: Find x - components of each vector

For $\vec{A}$: $A_x = 72\sin(32.0^{\circ})\approx 38.1$ m. For $\vec{B}$: $B_x=- 57.3\cos(36.0^{\circ})\approx - 46.4$ m. For $\vec{C}$: $C_x = 0$ m.

Step2: Find y - components of each vector

For $\vec{A}$: $A_y = 72\cos(32.0^{\circ})\approx 61.1$ m. For $\vec{B}$: $B_y=-57.3\sin(36.0^{\circ})\approx - 33.7$ m. For $\vec{C}$: $C_y=-17.8$ m.

Step3: Calculate the x - component of the resultant vector $\vec{R}$

$R_x=A_x + B_x+C_x=38.1-46.4 + 0=-8.3$ m.

Step4: Calculate the y - component of the resultant vector $\vec{R}$

$R_y=A_y + B_y+C_y=61.1-33.7-17.8 = 9.6$ m.

Step5: Calculate the magnitude of the resultant vector $\vec{R}$

$R=\sqrt{R_x^{2}+R_y^{2}}=\sqrt{(-8.3)^{2}+9.6^{2}}\approx12.7$ m.

Step6: Calculate the direction of the resultant vector $\vec{R}$

$\theta=\arctan\left(\frac{R_y}{R_x}\right)=\arctan\left(\frac{9.6}{-8.3}\right)\approx130.8^{\circ}$ counter - clockwise from the positive x - axis or $59.2^{\circ}$ north of west.

Answer:

The magnitude of the resultant displacement is approximately $12.7$ m and the direction is approximately $59.2^{\circ}$ north of west.