Answer to Question #213091 in Linear Algebra for Dhruv bartwal

Question

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

Answer 1

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

Proof

$(v_1+v_2)\cdot(v_2+v_3)=v_1\cdot{v_2}+v_1\cdot{v_3}+v_2\cdot{v_2}+v_2\cdot{v_3}\\$

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

$v_1\cdot v_2=v_2\cdot v_3=v_1\cdot v_3=0\\ (v_1+v_2)\cdot(v_2+v_3)=v_2\cdot v_2=\|v_2\|^2$

Alternatively

$(v_1+v_2)\cdot(v_3+v_1)=\|v_1\|^2\\ (v_2+v_3)\cdot(v_3+v_1)=\|v_3\|^2\\$

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