Calculating Vector Distance in R^n Using Dot Product - Homework Solution

[phrygian]

Homework Statement

Let x and y be vectos in R^n such that ||x|| = ||y|| = 1 and x^T * y = 0. Use Eq. (1) to show that ||x-y|| = sqrt(2).

Homework Equations

Eq. (1): ||x|| = sqrt(x^T * X)

The Attempt at a Solution

I could figure this out knowing that the dot product of the vectors is zero so the are perpendicular, the two sides of the triangle are 1 so the distance is sqrt(2) but this problem wants the answer to be found with Eq. (1) and I don't know how to do that and how to find ||x-y||.

Thanks for the help

 

[HallsofIvy]

Homework Statement

Let x and y be vectos in R^n such that ||x|| = ||y|| = 1 and x^T * y = 0. Use Eq. (1) to show that ||x-y|| = sqrt(2).

Homework Equations

Eq. (1): ||x|| = sqrt(x^T * X)

The Attempt at a Solution

I could figure this out knowing that the dot product of the vectors is zero so the are perpendicular, the two sides of the triangle are 1 so the distance is sqrt(2) but this problem wants the answer to be found with Eq. (1) and I don't know how to do that and how to find ||x-y||.

Thanks for the help

Your formula says that ||x- y||= \sqrt{(x- y)^T(x- y)}= \sqrt{x^Tx- x^Ty- y^Tx+ y^Ty}. Both x^Ty and y^Tx are 0.