Inequalities

Step 1: Vector addition and magnitude.
We are given two non-zero vectors and , and we need to prove the inequality .
This inequality is known as the **Triangle Inequality** for vectors.
Step 2: Proof of the Triangle Inequality.
We know that the square of the magnitude of a vector is equal to the dot product of the vector with itself. Therefore, we have: Expanding the dot product: This simplifies to: Step 3: Using the Cauchy-Schwarz inequality.
From the Cauchy-Schwarz inequality, we know that: Thus: Step 4: Taking square roots.
Taking the square root of both sides, we get: Since , we have: Step 5: Conclusion.
Thus, the inequality is proven, and it is known as the **Triangle Inequality** for vectors.