Understanding Vector Inequalities

This question asks us to evaluate the correctness of two common inequalities involving vectors. Vectors are mathematical objects that have both magnitude (or length) and direction. The magnitude of a vector \({\rm{\vec a}}\) is denoted by \(|\vec{a}|\).

Analyzing Inequality 1: The Triangle Inequality for Vectors

The first inequality is \( \left| {{\rm{\vec a}} + {\rm{\vec b}}} \right| \le \left| {{\rm{\vec a}}} \right| + \left| {{\rm{\vec b}}} \right| \).

• This is known as the Triangle Inequality for vectors.
• Geometrically, if we represent vectors \({\rm{\vec a}}\) and \({\rm{\vec b}}\) as two sides of a triangle, their sum \({\rm{\vec a}} + {\rm{\vec b}}\) represents the third side.
• The inequality states that the length of one side of a triangle (\(|{\rm{\vec a}} + {\rm{\vec b}}|\)) is less than or equal to the sum of the lengths of the other two sides (\(|{\rm{\vec a}}|\) + \(|{\rm{\vec b}}|\)). This is a fundamental geometric principle.
• Algebraically, we can prove this using the dot product. We know that \(|\vec{v}|^2 = \vec{v} \cdot \vec{v}\). $$|\vec{a} + \vec{b}|^2 = (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b})$$ $$|\vec{a} + \vec{b}|^2 = \vec{a} \cdot \vec{a} + \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{b}$$ Since the dot product is commutative (\(\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}\)), $$|\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2(\vec{a} \cdot \vec{b})$$ We know that \( \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta \), where \(\theta\) is the angle between \({\rm{\vec a}}\) and \({\rm{\vec b}}\). Since \(\cos\theta \le 1\), we have \( \vec{a} \cdot \vec{b} \le |\vec{a}||\vec{b}| \). Substituting this into the equation: $$|\vec{a} + \vec{b}|^2 \le |\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}|$$ $$|\vec{a} + \vec{b}|^2 \le (|\vec{a}| + |\vec{b}|)^2$$ Taking the square root of both sides (since magnitude is non-negative): $$|\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}|$$
• The inequality holds true for all vectors \({\rm{\vec a}}\) and \({\rm{\vec b}}\). Thus, Inequality 1 is correct.

Analyzing Inequality 2: The Reverse Triangle Inequality (Difference)

The second inequality is \( \left| {{\rm{\vec a}} - {\rm{\vec b}}} \right| \ge \left| {{\rm{\vec a}}} \right| - \left| {{\rm{\vec b}}} \right| \).

• This inequality is a variation of the triangle inequality, sometimes called the reverse triangle inequality or the difference inequality.
• We can derive this from the standard triangle inequality \(|\vec{x} + \vec{y}| \le |\vec{x}| + |\vec{y}|\). Let \( \vec{x} = \vec{a} - \vec{b} \) and \( \vec{y} = \vec{b} \). Then \( \vec{x} + \vec{y} = (\vec{a} - \vec{b}) + \vec{b} = \vec{a} \). Applying the triangle inequality: $$|\vec{a}| \le |\vec{a} - \vec{b}| + |\vec{b}|$$ Rearranging the terms, we get: $$|\vec{a}| - |\vec{b}| \le |\vec{a} - \vec{b}|$$ This is exactly the inequality given as Inequality 2.
• We can also derive it by swapping the roles of \({\rm{\vec a}}\) and \({\rm{\vec b}}\): $$|\vec{b}| = |(\vec{b} - \vec{a}) + \vec{a}| \le |\vec{b} - \vec{a}| + |\vec{a}|$$ $$|\vec{b}| - |\vec{a}| \le |\vec{b} - \vec{a}|$$ Since \( |\vec{b} - \vec{a}| = |-(\vec{a} - \vec{b})| = |\vec{a} - \vec{b}| \), we have: $$|\vec{b}| - |\vec{a}| \le |\vec{a} - \vec{b}|$$ This can be written as \( -(|\vec{a}| - |\vec{b}|) \le |\vec{a} - \vec{b}| \).
• Combining the two results, \(|\vec{a}| - |\vec{b}| \le |\vec{a} - \vec{b}|\) and \( -(|\vec{a}| - |\vec{b}|) \le |\vec{a} - \vec{b}|\), gives us \( ||{\rm{\vec a}}| - |{\rm{\vec b}}|| \le |{\rm{\vec a}} - {\rm{\vec b}}| \). The given inequality \( \left| {{\rm{\vec a}} - {\rm{\vec b}}} \right| \ge \left| {{\rm{\vec a}}} \right| - \left| {{\rm{\vec b}}} \right| \) is indeed one part of this more general inequality, and it is always true.
• Thus, Inequality 2 is correct.

Conclusion

Both Inequality 1 (\( \left| {{\rm{\vec a}} + {\rm{\vec b}}} \right| \le \left| {{\rm{\vec a}}} \right| + \left| {{\rm{\vec b}}} \right| \)) and Inequality 2 (\( \left| {{\rm{\vec a}} - {\rm{\vec b}}} \right| \ge \left| {{\rm{\vec a}}} \right| - \left| {{\rm{\vec b}}} \right| \)) are correct vector inequalities.

Based on the analysis, both inequality 1 and inequality 2 are correct.

Revision Table: Key Vector Concepts

Additional Information on Vector Inequalities

Vector inequalities are crucial in many areas of mathematics and physics. They establish relationships between the magnitudes of vectors and their sums or differences. These inequalities are derived from the fundamental properties of vector spaces and norms (which generalize the concept of magnitude).

• The triangle inequality, \(|\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}|\), signifies that taking a direct path (the vector sum) is always shorter than or equal to taking an indirect path (summing the individual vector lengths). Equality holds when vectors \({\rm{\vec a}}\) and \({\rm{\vec b}}\) are in the same direction.
• The reverse triangle inequality, \(||{\rm{\vec a}}| - |{\rm{\vec b}}|| \le |{\rm{\vec a}} - {\rm{\vec b}}|\), provides a lower bound for the magnitude of the difference of two vectors. Equality holds when vectors \({\rm{\vec a}}\) and \({\rm{\vec b}}\) are in the same direction (for \(|{\rm{\vec a}}| \ge |{\rm{\vec b}}|\)) or opposite directions. The specific inequality \( \left| {{\rm{\vec a}} - {\rm{\vec b}}} \right| \ge \left| {{\rm{\vec a}}} \right| - \left| {{\rm{\vec b}}} \right| \) is always true because the magnitude on the left is always non-negative, and the difference on the right could be negative or positive. However, even if the difference is positive, the inequality holds due to the derivation from the standard triangle inequality as shown above.
• These inequalities are essential in defining norms in abstract vector spaces and play a vital role in functional analysis, geometry, and optimization.