Suppose that u = and v = are vectors such that | u+v |^2 = | u |^2 + | v |^2. Prove that u and v are orthogonal.

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Mathematics

Gillian Hodges

12 September, 09:33

Suppose that u = and v = are vectors such that | u+v |^2 = | u |^2 + | v |^2. Prove that u and v are orthogonal.

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Lucy Lyons · Answer 1

If u = (u1, u2, u3) andv = (v1, v2, v3), then the dot product of u and v is u·v=u1v1+u2v2+u3v3. For instance, the dot product of u=i-2j-3kandv = 2j-kisu·v = 1·0 + (-2) ·2 + (-3) (-1) = -1.

Properties of the Dot Product.

Let u, v, and w be three vectors and let c be a real number. Then u·v=v·u, (u+v) ·w=u·w+v·w, (cu) ·v=c (u·v).

Further, u·u=|u|2.

Thus, if u=0is the zerovector, then u·u = 0, and otherwise u·u>0.1

Orthogonality Two vectors u and v are said to be orthogonal (perpendicular), if the angle between them is 90◦. Theorem. Two vectors u and v are orthogonal if and only if u·v = 0.