## University Calculus: Early Transcendentals (3rd Edition)

Since, we have two vectors $v_1$ and $v_2$ and $(v_1+v_2) \cdot (v_1-v_2)=v_1 \cdot v_1+v_2 \cdot v_1-v_1 \cdot v_2 -v_2 \cdot v_2$ Now, $v_1 \cdot v_1+v_2 \cdot v_1-v_1 \cdot v_2 -v_2 \cdot v_2=|v_1|^2-|v_2|^2$ The answer will be zero when $|v_1|^2 =|v_2|^2$, which is possible if both vectors have the same length. Thus, $|v_1|^2-|v_2|^2=|v_1|^2-|v_1|^2=0$ Hence, the sum of two vectors having the same length will always be orthogonal to their difference.