A boat sails 20 km in a direction of N75°E. Draw the 20 km travelled in a direction of
N75°E as a vector v in standard position. Find the component form of the vector v. Round your
answers to the nearest hundredth.
To draw the vector in standard position, we need to visualize a coordinate plane where the positive x-axis represents east and the positive y-axis represents north.
First, draw the x-axis (east-west axis) and the y-axis (north-south axis) intersecting at the origin (0, 0).
To draw the vector v in the direction of N75°E, start at the origin and move 20 units in a direction 75° east of north.
Begin by moving 20 units up along the positive y-axis (direction of north) until you reach the point (0, 20). Then, from this point, rotate 75° clockwise and move along the positive x-axis (direction of east) for 20 units.
To find the component form of vector v, we need to break down the vector into its x-component and y-component.
The x-component is the horizontal displacement or length along the x-axis. To calculate it, we use the trigonometric function cosine (cos). The formula is:
x-component = magnitude of the vector * cos(angle).
In this case, the magnitude of the vector is 20 km and the angle is 75°. Plugging these values into the formula, we get:
x-component = 20 km * cos(75°).
Using a calculator, evaluate this expression to find the x-component.
Next, to find the y-component, we use the trigonometric function sine (sin). The formula is:
y-component = magnitude of the vector * sin(angle).
Again, using the values of the magnitude (20 km) and angle (75°), plug them into the formula to find the y-component:
y-component = 20 km * sin(75°).
Evaluate this expression using a calculator to find the y-component.
Round your answers to the nearest hundredth as instructed.