a boat starts at point A, moves 3km due north, then 2 km due east, then 1 km due south, and then 4 km due east to point b. Find the distance AB
Add up each component (north or east) separately, and finally find the distance by the vectorial sum (using Pythagoras).
North: 3-1=2km
East: 2+4=6
Distance = √(2²+6²)
=√(40)
=2√(10)
I don't understand where for North you put 3-1. The boat moved 1km South and 3km North. Is that how this was done?
In vectors, directions are signed.
This means that if we take the positive direction as north, then going south is backwards, and is thus represented as negative. Otherwise it would not make a difference between going 3 km north and 1 km north (3+1), and going 3 km north and 1 km south (3-1).
Hope that clarifies how it's done.
similar to other
To find the distance AB, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the boat moves north, east, south, and then east again, forming two right-angled triangles. Let's name the points where the boat changes its direction:
- Point A: starting point
- Point N: after moving 3 km due north
- Point E: after moving 2 km due east from point N
- Point S: after moving 1 km due south from point E
- Point B: after moving 4 km due east from point S
Now, let's calculate the distances between the points:
1. The distance between point A and point N is 3 km (as given in the question).
2. The distance between point N and point E is 2 km (as given in the question).
3. The distance between point E and point S is the difference between their latitudes, which is 3 km (3 km north and then 1 km south).
4. The distance between point S and point B is 4 km (as given in the question).
Now, let's calculate the distance AB:
- The distance AB is the sum of the distances AE and EB.
The distance AE is the hypotenuse of the right-angled triangle, with sides NE and EA. We can calculate it using the Pythagorean theorem:
AE^2 = NE^2 + EA^2
AE^2 = (3 km)^2 + (2 km)^2
AE^2 = 9 km^2 + 4 km^2
AE^2 = 13 km^2
Taking the square root of both sides:
AE = sqrt(13) km
The distance EB is simply 4 km (as given in the question).
Therefore, the distance AB is:
AB = AE + EB
AB = sqrt(13) km + 4 km
Calculating the numerical value:
AB ≈ 3.61 km + 4 km
AB ≈ 7.61 km
Hence, the distance AB is approximately 7.61 km.