If $\|a \vec{i}+b \vec{j}\|\|c \vec{i}+d \vec{j}\|$ , then a b and c d.

Question

If $\|a \vec{i}+b \vec{j}\|>\|c \vec{i}+d \vec{j}\|$ , then a > b and c > d.

Quizplus · Accepted Answer

The magnitude of a vector $\|a \vec{i}+b \vec{j}\|$ is determined by $\sqrt{a^2 + b^2}$, which means the inequality $\|a \vec{i}+b \vec{j}\|>\|c \vec{i}+d \vec{j}\|$ only implies that the sum of the squares of the components of the first vector is greater than that of the second vector. It does not imply anything about the individual relationships between $a$ and $b$ or $c$ and $d$.

If ∥ai⃗+bj⃗∥>∥ci⃗+dj⃗∥\|a \vec{i}+b \vec{j}\|>\|c \vec{i}+d \vec{j}\|∥ai+bj​∥>∥ci+dj​∥ , Then a > B and C > D

If $\|a \vec{i}+b \vec{j}\|>\|c \vec{i}+d \vec{j}\|$ , Then a > B and C > D