Pythagorean Theorem: Proving Right-Angled Triangle OAB with OA=i+j+k, OB=2i-2j

To show that triangle OAB is right-angled, we need to prove that the square of the length of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides.
\nGiven:
OA = i + j + k
OB = 2i - 2j
\nFirst, let's calculate the lengths of OA and OB using the distance formula:
OA = √(i^2 + j^2 + k^2)
OB = √((2i)^2 + (-2j)^2)
\nSimplifying these expressions, we get:
OA = √(i^2 + j^2 + k^2)
OB = √(4i^2 + 4j^2)
\nNow, let's calculate the square of the lengths of OA and OB:
OA^2 = (i^2 + j^2 + k^2)^2
OB^2 = (4i^2 + 4j^2)^2
\nNow, let's calculate the sum of the squares of OA and OB:
OA^2 + OB^2 = (i^2 + j^2 + k^2)^2 + (4i^2 + 4j^2)^2
\nUsing the Pythagorean theorem, if OA^2 + OB^2 = OC^2, where OC is the length of the remaining side, then triangle OAB is right-angled.
\nSo, if we can prove that OA^2 + OB^2 = OC^2, then triangle OAB is right-angled.
\nHowever, without additional information about the length of OC or any other sides, we cannot determine if triangle OAB is right-angled using only the given information.