Question #213091

If { v1, v2, v3} is a set of mutually orthogonal vector, then so is { v1+v2, v2+v2,v3+v1}

True or false with full explanation


1
Expert's answer
2021-07-06T10:18:37-0400

The statement is false, {v1+v2,v2+v3,v3+v1} is not a set of mutually orthogonal vectors.

Proof

(v1+v2)(v2+v3)=v1v2+v1v3+v2v2+v2v3(v_1+v_2)\cdot(v_2+v_3)=v_1\cdot{v_2}+v_1\cdot{v_3}+v_2\cdot{v_2}+v_2\cdot{v_3}\\

Since  { v1, v2, v3} is a set of mutually orthogonal vectors,

v1v2=v2v3=v1v3=0(v1+v2)(v2+v3)=v2v2=v22v_1\cdot v_2=v_2\cdot v_3=v_1\cdot v_3=0\\ (v_1+v_2)\cdot(v_2+v_3)=v_2\cdot v_2=\|v_2\|^2

Alternatively

(v1+v2)(v3+v1)=v12(v2+v3)(v3+v1)=v32(v_1+v_2)\cdot(v_3+v_1)=\|v_1\|^2\\ (v_2+v_3)\cdot(v_3+v_1)=\|v_3\|^2\\

The dot products of the set of vectors are not zero. Hence, they are not mutually orthogonal.


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