Given the points O = (0, 0, 0), A = (2, 0, -2) and B = (1, -1, 0) in 3-dimensional space, I need to find, with proof, the angles AOB and OBA, the lengths OB and AB and the area of the triangle OAB.
This is what I have done:
Determined what OA, AB and OB are.
OA = A - O
= (2, 0, -2) - (0, 0, 0)
= (2, 0, -2)
AB = B - A
= (1, -1, 0) - (2, 0, -2)
= (-1, 1, 2)
OB = B - O
= (1, -1, 0) - (0, 0, 0)
= (1, -1, 0)
To find the angle AOB, I used the formula cos^-1(||OA||^2 + ||OB||^2 - ||AB||^2 / 2 * ||OA|| * ||OB||) with the coordinates obtained to get the answer π/3.
To find the angle OBA, I used the formula cos(||AB||^2 + ||OB||^2 - ||OA||^2 / 2 * ||AB|| * ||OB||) with the coordinates obtained to get the answer π/2.
To find the lengths OB and AB, I used the formula |z| = √(x^2 + y^2 + z^2) with the coordinates obtained, so:
||OB|| = √(1^2 + (-1)^2 + 0^2) = √2
||AB|| = √(1^2 + 1^2 + 2^2) = √6
To find the area of the triangle OAB, I used the formula 1/2 * ||AB|| * ||OB|| with the lengths obtained, so:
Area = 1/2 * √6 * √2 = √3 unit^2.
Besides plotting the points, labelling angles etc. on a diagram, is the above sufficient to prove each part and are all the answers correct?