Find a unit vector in the direction of v = (1,-1). Write your answer as a linear combination of the standard unit vectors i and j. Round each component to the nearest hundredth, if necessary.
A 0.71i - 0.71j
B 0i - 0j
C 1.41i - 1.41j
D 1i - 1j
E 2.83i - 3.83j
F -1i - 1j
||v||^2 = 1^2 + 1^2 = 2
So, you need to divide v by sqrt(2) to normalize it to one.
To find a unit vector in the direction of v = (1,-1), you need to divide each component of v by the magnitude of v. The magnitude of v is the square root of the sum of the squares of its components, so for v = (1,-1), the magnitude is √(1^2 + (-1)^2) = √(1+1) = √2.
Next, divide each component of v by the magnitude √2:
v/√2 = (1/√2, -1/√2)
Now, we can write this as a linear combination of the standard unit vectors i and j:
(1/√2)i + (-1/√2)j
Rounding each component to the nearest hundredth, we get:
(0.71)i + (-0.71)j
So the correct answer is A: 0.71i - 0.71j.