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].
if w = [a,b]
then |w| = √(a^2 + b^2)
so |u| and |v| are easy
u - v =[3,-4] - [8,15]
= [-5,-19]
now apply the above formula again
the same rules apply to the 3-D vectors.
Post your answers so I can check them
will do. thanks
Oh, vectors huh? Don't worry, I got this!
For the first one:
|u| = √(3² + (-4)²) = √(9 + 16) = √25 = 5
|v| = √(8² + 15²) = √(64 + 225) = √289 = 17
Now, let's calculate |u - v|:
|u - v| = √((3 - 8)² + (-4 - 15)²) = √((-5)² + (-19)²) = √(25 + 361) = √386
And finally, let's calculate |u| - |v|:
|u| - |v| = 5 - 17 = -12
For the second one:
|u| = √(2² + 6² + (-3)²) = √(4 + 36 + 9) = √49 = 7
|v| = √(2² + 2² + 1²) = √(4 + 4 + 1) = √9 = 3
Now, let's calculate |u - v|:
|u - v| = √((2 - 2)² + (6 - 2)² + (-3 - 1)²) = √(0 + 16 + 16) = √32
And finally, let's calculate |u| - |v|:
|u| - |v| = 7 - 3 = 4
Hope all that math didn't give you a headache!
To calculate the magnitudes of vectors u and v, we use the formula:
|u| = sqrt(u1^2 + u2^2 + ... + un^2)
For vector u = [3, -4]:
|u| = sqrt(3^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5
For vector v = [8, 15]:
|v| = sqrt(8^2 + 15^2) = sqrt(64 + 225) = sqrt(289) = 17
To calculate the magnitude of vector u - v, we subtract the corresponding components and take the magnitude:
u - v = [3 - 8, -4 - 15] = [-5, -19]
|u - v| = sqrt((-5)^2 + (-19)^2) = sqrt(25 + 361) = sqrt(386) ≈ 19.65
To calculate |u| - |v|, we subtract the magnitudes:
|u| - |v| = 5 - 17 = -12
For vector u = [2, 6, -3]:
|u| = sqrt(2^2 + 6^2 + (-3)^2) = sqrt(4 + 36 + 9) = sqrt(49) = 7
For vector v = [2, 2, 1]:
|v| = sqrt(2^2 + 2^2 + 1^2) = sqrt(4 + 4 + 1) = sqrt(9) = 3
To calculate the magnitude of vector u - v, we subtract the corresponding components and take the magnitude:
u - v = [2 - 2, 6 - 2, -3 - 1] = [0, 4, -4]
|u - v| = sqrt(0^2 + 4^2 + (-4)^2) = sqrt(0 + 16 + 16) = sqrt(32) ≈ 5.66
To calculate |u| - |v|, we subtract the magnitudes:
|u| - |v| = 7 - 3 = 4
To calculate the values for |u|, |v|, |u - v|, and |u| - |v|, we need to use some basic vector operations.
1. Magnitude of a vector:
The magnitude or length of a vector can be calculated using the Euclidean norm formula, which is the square root of the sum of the squares of its components.
2. Vector subtraction:
To find the difference between two vectors, u - v, we subtract the corresponding components of the vectors.
Let's calculate the values step by step:
For the first set of vectors, u = [3, -4] and v = [8, 15]:
1. |u| (Magnitude of u):
- To find the magnitude of u, we use the Euclidean norm formula:
|u| = sqrt((3^2) + (-4^2)) = sqrt(9 + 16) = sqrt(25) = 5.
2. |v| (Magnitude of v):
- Using the same formula:
|v| = sqrt((8^2) + (15^2)) = sqrt(64 + 225) = sqrt(289) = 17.
3. |u - v| (Magnitude of u - v):
- Subtract the corresponding components of u and v:
u - v = [3 - 8, -4 - 15] = [-5, -19].
- Calculate the magnitude:
|u - v| = sqrt((-5^2) + (-19^2)) = sqrt(25 + 361) = sqrt(386).
4. |u| - |v| (Magnitude of u minus magnitude of v):
- Calculate the magnitudes first:
|u| = 5 and |v| = 17.
- Subtract the magnitudes:
|u| - |v| = 5 - 17 = -12.
Now, let's calculate the values for the second set of vectors, u = [2, 6, -3] and v = [2, 2, 1]:
1. |u| (Magnitude of u):
|u| = sqrt((2^2) + (6^2) + (-3^2)) = sqrt(4 + 36 + 9) = sqrt(49) = 7.
2. |v| (Magnitude of v):
|v| = sqrt((2^2) + (2^2) + (1^2)) = sqrt(4 + 4 + 1) = sqrt(9) = 3.
3. |u - v| (Magnitude of u - v):
u - v = [2 - 2, 6 - 2, -3 - 1] = [0, 4, -4].
|u - v| = sqrt((0^2) + (4^2) + (-4^2)) = sqrt(0 + 16 + 16) = sqrt(32).
4. |u| - |v| (Magnitude of u minus magnitude of v):
|u| - |v| = 7 - 3 = 4.
So, the values for the two sets of vectors are:
For u = [3, -4] and v = [8, 15]:
|u| = 5, |v| = 17, |u - v| = sqrt(386), and |u| - |v| = -12.
For u = [2, 6, -3] and v = [2, 2, 1]:
|u| = 7, |v| = 3, |u - v| = sqrt(32), and |u| - |v| = 4.