Find the vector v with the given magnitude and the same direction as u.
Magnitude: v=3 direction: u=(4,-4)
I am not sure what to do here or how to start.
Have you learned about unit vectors?
a unit vector (length = 1) in the same direction of
(a,b) = (a/√(a^2+b2) , b/√(a^2 + b^2))
so a unit vector along (4,-4) is (4/√32, -4/√32)
so one of length 3 is
3(4/√32, -4/√32)
= (12/√32, -12/√32)
which rationalizes to
(3√2/2 , -3√2/2)
To find a vector with the same direction as u but with a different magnitude, you can use the concept of unit vectors. A unit vector is a vector with a magnitude of 1 that points in the same direction as another vector.
To find the unit vector that has the same direction as u, you need to divide u by its magnitude. This will give you a vector of length 1 that points in the same direction as u.
First, calculate the magnitude of u using the formula:
|u| = √(x^2 + y^2)
where x and y are the components of u.
In this case, u = (4, -4), so:
|u| = √(4^2 + (-4)^2) = √(16 + 16) = √32 ≈ 5.657
Next, divide each component of u by its magnitude to find the unit vector:
v = u / |u| = (4 / 5.657, -4 / 5.657) = (0.707, -0.707)
Finally, to find the vector v with the given magnitude, simply multiply each component of the unit vector v by the desired magnitude:
v = 3 * (0.707, -0.707) = (2.121, -2.121).
Therefore, the vector v with a magnitude of 3 and the same direction as u = (4,-4) is v = (2.121, -2.121).