Answer: To find the angle between the directions of the two boats, we need to calculate the angle between sides AB and AC of the triangle ABC. Let's follow these steps:
Step 1: Calculate the vectors AB and AC.
The vector AB is the difference between the coordinates of points B and A:
AB = (3 - 1, 8 - 2) = (2, 6)
The vector AC is the difference between the coordinates of points C and A:
AC = (7 - 1, 4 - 2) = (6, 2)
Step 2: Calculate the dot product of vectors AB and AC.
The dot product of two vectors A and B is given by:
A · B = |A| * |B| * cos(theta)
where |A| and |B| are the magnitudes of vectors A and B, and theta is the angle between them.
In our case, since we want to find the angle between sides AB and AC:
AB · AC = |AB| * |AC| * cos(theta)
Step 3: Calculate the magnitudes of vectors AB and AC.
The magnitude of a vector (a, b) is given by:
|A| = sqrt(a^2 + b^2)
For AB:
|AB| = sqrt(2^2 + 6^2) = sqrt(40) = 2 * sqrt(10)
For AC:
|AC| = sqrt(6^2 + 2^2) = sqrt(40) = 2 * sqrt(10)
Step 4: Calculate the dot product of AB and AC.
AB · AC = (2, 6) · (6, 2) = 2 * 6 + 6 * 2 = 12 + 12 = 24
Step 5: Calculate the angle theta.
Now we can find theta using the formula:
AB · AC = |AB| * |AC| * cos(theta)
24 = 2 * sqrt(10) * 2 * sqrt(10) * cos(theta)
Divide both sides by 4 * 10:
cos(theta) = 24 / (4 * 10) = 6 / 10 = 0.6
Step 6: Find the angle theta.
Finally, we can find theta by taking the arccosine (inverse cosine) of 0.6:
theta = arccos(0.6) ≈ 53.13 degrees
So, the angle between the directions of the two boats is approximately 53.13 degrees.