A sailboat race course consists of four legs, defined by the displacement vectors , , , and , as the drawing indicates. The magnitudes of the first three vectors are A = 2.80 km, B = 4.70 km, and C = 4.50 km. The finish line of the course coincides with the starting line. Using the data in the drawing, find (a) the distance of the fourth leg and (b) the angle .
given the drawing as shown below, I calculate 3.62 km at 46°
To find the distance of the fourth leg, we need to use vector addition.
Given:
Magnitude of vector A = A = 2.80 km
Magnitude of vector B = B = 4.70 km
Magnitude of vector C = C = 4.50 km
(a) Distance of the fourth leg:
To find the distance of the fourth leg, we can add vectors A, B, and C together using vector addition.
The resultant vector of A and B is:
AB = A + B
The resultant vector of AB and C is:
ABC = AB + C
We can calculate the magnitude of ABC using the Pythagorean theorem:
|ABC|^2 = |AB|^2 + |C|^2
|ABC|^2 = (A + B)^2 + C^2
|ABC|^2 = (A^2 + 2AB + B^2) + C^2
|ABC|^2 = (2.80 km)^2 + 2(A)(B) + (4.70 km)^2 + (4.50 km)^2
|ABC|^2 = 7.84 km^2 + 2(2.80 km)(4.70 km) + 22.09 km^2 + 20.25 km^2
|ABC|^2 = 55.36 km^2 + 26.32 km^2 + 42.34 km^2 + 20.25 km^2
|ABC|^2 = 144.27 km^2
So, the magnitude of ABC is:
|ABC| = √(144.27 km^2)
|ABC| ≈ 12.01 km
Therefore, the distance of the fourth leg is approximately 12.01 km.
(b) Angle θ:
To find the angle θ, we can use the law of cosine.
In triangle ABC, the angle θ is opposite to side C.
Using the law of cosine, we can write:
C^2 = A^2 + B^2 - 2AB * cos(θ)
Substituting the given values:
(4.50 km)^2 = (2.80 km)^2 + (4.70 km)^2 - 2(2.80 km)(4.70 km) * cos(θ)
20.25 km^2 = 7.84 km^2 + 22.09 km^2 - 26.32 km^2 * cos(θ)
20.25 km^2 = 30.93 km^2 - 26.32 km^2 * cos(θ)
20.25 km^2 - 30.93 km^2 = -26.32 km^2 * cos(θ)
-10.68 km^2 = -26.32 km^2 * cos(θ)
cos(θ) = -10.68 km^2 / (-26.32 km^2)
cos(θ) ≈ 0.405
Since the angle is between 0 and 180 degrees, we can use the inverse cosine function to find θ:
θ ≈ acos(0.405)
θ ≈ 66.17 degrees
Therefore, the angle θ is approximately 66.17 degrees.
To find the distance of the fourth leg of the sailboat race course, we can draw a vector diagram using the given displacement vectors A, B, and C.
Step 1: Draw the vectors A, B, and C.
- Draw a vector A of magnitude 2.80 km in the desired direction.
- Draw a vector B of magnitude 4.70 km starting from the end of vector A in the desired direction.
- Draw a vector C of magnitude 4.50 km starting from the end of vector B in the desired direction.
Step 2: Connect the end of vector C to the start of vector A to complete the vector diagram.
- Draw a vector D starting from the end of vector C and pointing towards the start of vector A.
Step 3: Find the magnitude of vector D.
- The magnitude of vector D is the distance of the fourth leg of the sailboat race course.
Step 4: Use the vector diagram to find the angle θ.
- Measure the angle between the vectors A and D using a protractor or geometrically.
(a) To find the distance of the fourth leg (magnitude of vector D), you need to measure the length of vector D using a ruler or any other measuring tool on the vector diagram.
(b) To find the angle θ between vectors A and D, you need to measure the angle between these two vectors using a protractor or calculate it using trigonometry. You can use the information provided in the vector diagram to determine the angle.
Note: The given displacement vectors can be added together using the head-to-tail method to create the vector diagram accurately.