Let u = [3,−4] and v = [8, 15]. Calculate the four numbers |u|, |v|, |u − v|, and |u| − |v|. Do the same with the vectors u = [2, 6,−3] and v = [2, 2, 1].
To calculate the magnitude of a vector, denoted as |u|, we can use the formula:
|u| = √(u₁² + u₂²)
Let's calculate the four numbers for the first set of vectors, u = [3, -4] and v = [8, 15]:
1. Magnitude of vector u:
|u| = √(3² + (-4)²)
= √(9 + 16)
= √25
= 5
2. Magnitude of vector v:
|v| = √(8² + 15²)
= √(64 + 225)
= √289
= 17
3. Magnitude of the difference of u and v:
|u - v| = √((3 - 8)² + (-4 - 15)²)
= √((-5)² + (-19)²)
= √(25 + 361)
= √386
≈ 19.65
4. Difference of magnitudes of u and v:
|u| - |v| = 5 - 17
= -12
Now, let's calculate the same for the second set of vectors, u = [2, 6, -3] and v = [2, 2, 1]:
1. Magnitude of vector u:
|u| = √(2² + 6² + (-3)²)
= √(4 + 36 + 9)
= √49
= 7
2. Magnitude of vector v:
|v| = √(2² + 2² + 1²)
= √(4 + 4 + 1)
= √9
= 3
3. Magnitude of the difference of u and v:
|u - v| = √((2 - 2)² + (6 - 2)² + (-3 - 1)²)
= √(0 + 16 + 16)
= √32
≈ 5.66
4. Difference of magnitudes of u and v:
|u| - |v| = 7 - 3
= 4
So, the four numbers for the first set of vectors are: |u| = 5, |v| = 17, |u - v| ≈ 19.65, |u| - |v| = -12.
And the four numbers for the second set of vectors are: |u| = 7, |v| = 3, |u - v| ≈ 5.66, |u| - |v| = 4.