Given u = 1/SQRT(10) (-2,1,-1,2). Find a unit vector v satisfying the condition that u . v = -1.

since u.v = |u||v|cosθ we must have cosθ = -1. In other words,

v = -u

Thanks for that Steve, it makes sense now. Just a follow up question, what kind of geometrical objects that has cosθ = -1?

To find a unit vector v that satisfies the condition u . v = -1, we can follow these steps:

1. Calculate the magnitude of u, denoted as |u|:
- The magnitude of u, denoted as |u|, is given by |u| = SQRT(u1^2 + u2^2 + u3^2 + u4^2).
- Plugging in the values of u, we have |u| = SQRT((-2/SQRT(10))^2 + (1/SQRT(10))^2 + (-1/SQRT(10))^2 + (2/SQRT(10))^2).
- Simplifying this expression, |u| = SQRT(4/10 + 1/10 + 1/10 + 4/10) = SQRT(10/10) = 1.

2. Define a unit vector v:
- A unit vector v is a vector that has a magnitude of 1. So, we have |v| = 1.

3. Solve for v:
- We will solve for v by setting up the dot product equation: u . v = -1.
- The dot product of u and v is given by u1*v1 + u2*v2 + u3*v3 + u4*v4.
- Plugging in the values of u and v, we have:
(-2/SQRT(10))v1 + (1/SQRT(10))v2 + (-1/SQRT(10))v3 + (2/SQRT(10))v4 = -1.
- Since |u| = 1 (as calculated earlier), we can divide both sides of the equation by √10:
-2v1 + v2 - v3 + 2v4 = -√10.

4. Choose values for v1, v2, v3, and v4:
- To satisfy the equation -2v1 + v2 - v3 + 2v4 = -√10, you can arbitrarily choose values for three of the variables (e.g., v1, v2, and v3).
- Then, you can solve for the remaining variable (e.g., v4).

5. Compute v:
- Once you have chosen values for v1, v2, v3, and v4, you can form the vector v using these values.
- Since |v| = 1 (as defined earlier), you can normalize the vector v by dividing each component by its magnitude.

Following these steps, you should be able to find a unit vector v that satisfies the condition u . v = -1.