Enaj goes out for a run and jogs $3.2 \mathrm{~km}$ at $20^{\circ} \mathrm{S}$ of E, then $2.7 \mathrm{~km}$ at $35^{\circ} \mathrm{W}$ of S and finally $4.5 \mathrm{~km}$ at $124^{\circ}$. What vector does she need to reach home? SOLVE Graphically and Mathematically
Answer
\(\boxed{\text{Enaj needs to jog 4.34 km at an angle of 18.3 degrees south of west to reach home.}}\)
Step 1 ：Convert each vector to Cartesian coordinates. The x-component is the length times the cosine of the angle, and the y-component is the length times the sine of the angle. Remember to use the correct signs for the angles.
Step 2 ：The first vector is \((3.2 \cos 70, -3.2 \sin 70) = (1.09, -2.97)\).
Step 3 ：The second vector is \((-2.7 \sin 55, -2.7 \cos 55) = (-2.21, -2.07)\).
Step 4 ：The third vector is \((4.5 \cos 124, 4.5 \sin 124) = (-2.97, 3.68)\).
Step 5 ：Add the x-components and y-components separately to get the total displacement in x and y.
Step 6 ：The total displacement is \((1.09 - 2.21 - 2.97, -2.97 - 2.07 + 3.68) = (-4.09, -1.36)\).
Step 7 ：The length of the vector Enaj needs to reach home is the square root of the sum of the squares of the x and y displacements.
Step 8 ：The length of the vector is \(\sqrt{(-4.09)^2 + (-1.36)^2} = 4.34 \text{ km}\).
Step 9 ：The angle of this vector is the arctangent of the y displacement divided by the x displacement. Use the correct quadrant for the angle.
Step 10 ：The angle of this vector is \(\arctan(-1.36 / -4.09) = 18.3 \text{ degrees}\). Since the displacement is to the west and south, this is 18.3 degrees south of west.
Step 11 ：So, Enaj needs to jog 4.34 km at an angle of 18.3 degrees south of west to reach home.
Step 12 ：\(\boxed{\text{Enaj needs to jog 4.34 km at an angle of 18.3 degrees south of west to reach home.}}\)
