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They are u and v two vectors. Show that u + v and u − v are perpendicular

**Function of several variables** Are u and v two vectors such that form an angle of 450 and ‖ u ‖ = ‖ v ‖ = 2. a) What is the modulus of u + v? And what about the u − v? b) Show that u + v and u − v are perpendicular.

0voto

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I had confused the title of your question, which is not always the u+v is perpendicular to u-v; yes is given in this particular case, with the same module. If the angle of u; v is 45°, and the modulus of the two is 2: By parallelogram (Theorem and the cosine): |u+v|^2 = u^2+v^2 - 2uvCos(180|-45°); remember that the sum is the diagonal, the greater of the parallelogram. |u+v| = √ \[4+4- 2\*2\*2\*(-√2/2) \]; |u+v| = 3.69; |u-v| = √ (u^2+v^2 - 2uvCos45°); that is, the diagonal minor of the parallelogram. |u-v| = 2.34 We can see that this parallelogram has four equal sides, that is to say that it is a rhombus, and as such, both diagonals intersect orthogonally, and this shows that (diagonal greater) is orthogonal to (diagonal minor).

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