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.
2 pts. B. Next, the boat turns and sails 10 km in a direction of S60°E. Draw the next part of the
boat’s voyage as a vector w in standard position. Find the component form of the vector w.
Round your answers to the nearest hundredth.
6 pts. C. Draw the vector sum v + w, which represents the distance the boat is from her starting
place. Find the component form of v + w, the magnitude of v + w, and the direction.
I can't help with the drawing, so if that's a big problem, review your text and google stuff.
Assuming that you drew vector v, then it should be clear that the components are
x: 20 sin70° = 18.79
y: 20 cos70° = 6.84
similarly, w has components
x: 10 sin60° = 8.66
y: -10 cos60° = -5.00
To get v+w, just add the respective components
w = <27.45,1.84>
|w| = √(27.45^2 + 1.84^2) = 27.51
θ = 3.83°
As a heading, that is E3.83°N or 86.17° on the 360° from due North orientation.
To draw the vector v in standard position, start from the origin and move 20 km in a direction of N75°E.
To find the component form of the vector v, we can break it down into its horizontal and vertical components.
The horizontal component can be found by multiplying the magnitude (20 km) by the cosine of the angle (75°):
Horizontal component of v = 20 km * cos(75°) ≈ 5.26 km
The vertical component can be found by multiplying the magnitude (20 km) by the sine of the angle (75°):
Vertical component of v = 20 km * sin(75°) ≈ 19.34 km
Therefore, the component form of vector v is approximately (5.26 km, 19.34 km).
Now let's move on to vector w.
To draw the vector w in standard position, start from the endpoint of vector v and move 10 km in a direction of S60°E.
To find the component form of the vector w, we can again break it down into its horizontal and vertical components.
The horizontal component can be found by multiplying the magnitude (10 km) by the cosine of the angle (60°):
Horizontal component of w = 10 km * cos(60°) ≈ 5 km
The vertical component can be found by multiplying the magnitude (10 km) by the sine of the angle (60°):
Vertical component of w = 10 km * sin(60°) ≈ 8.66 km
Therefore, the component form of vector w is approximately (5 km, 8.66 km).
Lastly, let's calculate the vector sum v + w.
To draw the vector sum v + w, simply connect the initial point of vector v with the terminal point of vector w.
To find the component form of v + w, add the corresponding components of v and w:
Horizontal component of v + w = 5.26 km + 5 km ≈ 10.26 km
Vertical component of v + w = 19.34 km + 8.66 km ≈ 2.99 km
Therefore, the component form of vector v + w is approximately (10.26 km, 2.99 km).
To find the magnitude of v + w, use the Pythagorean theorem:
Magnitude of v + w = sqrt((10.26 km)^2 + (2.99 km)^2) ≈ 10.71 km
To find the direction of v + w, use the inverse tangent function:
Direction of v + w = tan^(-1)(2.99 km / 10.26 km) ≈ 16.67°
Therefore, the magnitude of v + w is approximately 10.71 km, and the direction is approximately N16.67°E.
To draw the vectors as mentioned and find their component forms, follow these steps:
Step 1: Draw vector v in standard position
- Start by drawing a reference line, which represents the x-axis.
- From the origin of the x-axis, draw a line at an angle of 75° with respect to the north direction (upwards).
- Measure a length of 20 km along this line and mark the endpoint. This represents vector v.
Step 2: Find the component form of vector v
To find the component form of vector v, we need to determine the horizontal (x-component) and vertical (y-component) dimensions.
- The horizontal component (x-component) is calculated using the formula v_x = v * cos(θ), where v is the magnitude of vector v and θ is the angle it makes with the x-axis.
- The vertical component (y-component) is calculated using the formula v_y = v * sin(θ), where v is the magnitude of vector v and θ is the angle it makes with the x-axis.
In this case, v = 20 km, and the angle θ with the x-axis is 75°.
v_x = 20 * cos(75°)
v_y = 20 * sin(75°)
When rounded to the nearest hundredth, vector v can be represented by the component form (v_x, v_y).
Step 3: Draw vector w in standard position
- This time, take the endpoint of vector v as the starting point.
- From this point, draw a line at an angle of 60° below the x-axis.
- Measure a length of 10 km along this line and mark the endpoint. This represents vector w.
Step 4: Find the component form of vector w
Similar to finding the component form of vector v, we'll use the formulas for the horizontal and vertical components.
- The horizontal component (x-component) of w is calculated using the formula w_x = w * cos(θ), where w is the magnitude of vector w, and θ is the angle it makes with the x-axis.
- The vertical component (y-component) of w is calculated using the formula w_y = -w * sin(θ), where w is the magnitude of vector w, and θ is the angle it makes with the x-axis.
In this case, w = 10 km, and the angle θ is 60°.
w_x = 10 * cos(60°)
w_y = -10 * sin(60°)
When rounded to the nearest hundredth, vector w can be represented by the component form (w_x, w_y).
Step 5: Draw vector v + w
To draw the vector sum v + w, we need to add the respective horizontal and vertical components.
- Start by placing the initial point of vector v as the starting point for v + w.
- From this point, move the endpoint by adding the horizontal components (v_x + w_x) and vertical components (v_y + w_y).
Step 6: Find the component form of v + w
To find the component form of v + w, sum the respective horizontal and vertical components.
(v + w)_x = v_x + w_x
(v + w)_y = v_y + w_y
When rounded to the nearest hundredth, vector v + w can be represented by the component form ((v + w)_x, (v + w)_y).
Step 7: Find the magnitude and direction of v + w
The magnitude of v + w can be calculated using the formula ||v + w|| = sqrt((v + w)_x^2 + (v + w)_y^2).
The direction of v + w can be found using the formula θ = atan2((v + w)_y, (v + w)_x), where atan2 is the inverse tangent function that considers the signs of the components to determine the correct quadrant.
This completes the process of drawing the vectors and finding their component forms, magnitude, and direction.