Question

a helicopter flies 62 miles east, then turns and flies at an angle of 45° north of east for 199 miles. what is the magnitude of the helicopters resultant vector? draw a vector diagram. |\overrightarrow{r}| = ? miles round your answer to the nearest hundredth.

Explanation:

Step1: Analyze the components of each vector

The first vector (eastward) has magnitude \( 62 \) miles, so its east component is \( 62 \) miles and north component is \( 0 \) miles.
The second vector has magnitude \( 199 \) miles at an angle of \( 45^\circ \) north of east. Using trigonometry, the east component is \( 199\cos(45^\circ) \) and the north component is \( 199\sin(45^\circ) \).
\(\cos(45^\circ)=\sin(45^\circ)=\frac{\sqrt{2}}{2}\approx0.7071\)
East component of second vector: \( 199\times0.7071\approx140.7129 \)
North component of second vector: \( 199\times0.7071\approx140.7129 \)

Step2: Find the total east and north components

Total east component (\( R_x \)): \( 62 + 140.7129 = 202.7129 \)
Total north component (\( R_y \)): \( 0 + 140.7129 = 140.7129 \)

Step3: Calculate the magnitude of the resultant vector

Using the Pythagorean theorem, the magnitude of the resultant vector \( |\vec{R}|=\sqrt{R_x^2 + R_y^2} \)
\( R_x = 202.7129 \), \( R_y = 140.7129 \)
\( |\vec{R}|=\sqrt{(202.7129)^2+(140.7129)^2} \)
First, calculate \( (202.7129)^2\approx202.7129\times202.7129\approx41092.52 \)
\( (140.7129)^2\approx140.7129\times140.7129\approx19800.12 \)
Sum: \( 41092.52 + 19800.12 = 60892.64 \)
Then, \( |\vec{R}|=\sqrt{60892.64}\approx246.76 \)

Answer:

\( 246.76 \)