Problem 95 - College Algebra and Calculus: An Applied Approach

Question

Think About It \(\quad\) Consider \(|u+v|\) and \(|u|+|v|\). (a) Are the values of the expressions always equal? If not, under what conditions are they unequal? (b) If the two expressions are not equal for certain values of \(u\) and \(v\), is one of the expressions always greater than the other? Explain

Step-by-Step Solution

Answer

The expressions \(|u+v|\) and \(|u|+|v|\) are only always equal if the vectors \(u\) and \(v\) point in the exact same direction. If not, \(|u|+|v|\) is always greater than \(|u+v|\).

Step 1: Evaluate inequality

Consider two cases, case 1: when vectors \(u\) and \(v\) are not in the same direction and case 2: when vectors \(u\) and \(v\) are in the same direction. Using the triangle inequality, it can be shown that \(|u+v|\leq|u|+|v|\), so the values of the two expressions are not always equal.

Step 2: Determine conditions for inequality

For the expressions to be unequal, vectors \(u\) and \(v\) would need to be not pointing in the exact same direction. This happens because the sum of vectors in the same direction results in a magnitude that is equal to the sum of their magnitudes. In contrast, when the vectors are not in the same direction, the magnitude of their sum is less than the sum of their magnitudes.

Step 3: Compare the expressions

If the two expressions are not equal for certain values of \(u\) and \(v\), one of the expressions is always greater than the other. This is always the \(|u|+|v|\) expression, due to the triangle inequality.

Step 4: Conclusion

The values of the expressions can be equal only if the vectors are in the same direction, following the triangle inequality, \(|u+v|\) will always be less than or equal to \(|u|+|v|\).

Key Concepts

Vector Addition

Vector addition is a process that combines two or more vectors to result in a new vector. This is performed by adding the corresponding components from each vector, effectively combining their magnitudes and directions. Think of vectors as arrows pointing in specific directions: when you add them, you're essentially joining them end-to-end to find the resultant arrow.

When vectors point in the same direction, their magnitudes add up, and the resultant vector is simply a straight addition of their lengths. However, when vectors are in different directions, the resultant vector's length is found using the parallelogram law, or by aligning them head-to-tail and creating a triangle or polygon.

• When vectors are in the same direction: \( |\mathbf{u} + \mathbf{v}| = |\mathbf{u}| + |\mathbf{v}| \)
• When vectors point in different directions, the magnitude of their sum is less than the sum of their magnitudes due to the angle between them.

The concept of vector addition is foundational in understanding how vectors interact, especially in physics and engineering where forces and velocities are involved. This ties directly into the triangle inequality concept, as it stipulates how these magnitudes relate depending on direction.

Magnitude of Vectors

The magnitude of a vector, often intuitive as its "length," is a measure of its size without considering its direction. In the context of the exercise, we look at magnitudes to compare different expressions like \(|\mathbf{u}+\mathbf{v}|\) and \(|\mathbf{u}| + |\mathbf{v}|\).

The magnitude of a vector \(\mathbf{u}\) with components \( (x,y) \) is calculated using the formula:\[|\mathbf{u}| = \sqrt{x^2 + y^2}\]

Understanding magnitudes helps us recognize how much of a physical quantity (like force or speed) a vector represents, independently of where it points. This simplifies many complex systems into simpler calculations. In relation to inequalities, magnitude plays a crucial role in determining if one expression exceeds another, highlighting their relative sizes and directions.

• Understanding the individual magnitudes of vectors helps in finding the resultant magnitude more accurately.
• When comparing magnitudes, ensure vectors are properly aligned or accounted for in their angles.

Geometric Interpretation of Inequalities

The geometric interpretation of inequalities provides a visual understanding of how distances and magnitudes relate, especially in the case of vector addition and the triangle inequality. The triangle inequality states that for any triangle, the length of one side must be less than or equal to the sum of the other two sides.

Translated into vector terms, this becomes:\[|\mathbf{u} + \mathbf{v}| \leq |\mathbf{u}| + |\mathbf{v}|\]
In simple terms, the direct path \(|\mathbf{u} + \mathbf{v}|\) is always the shortest path between two points.

• If vectors \(\mathbf{u}\) and \(\mathbf{v}\) are aligned, their combined magnitude equals the sum of their magnitudes.
• If they form an angle between them, the triangle formed illustrates the inequality, showing the direct path is less than or equal to the sum of the detoured paths.

Understanding these geometric interpretations aids in comprehending why some inequalities hold, as well as their implications in real-world contexts. This is crucial for students in visualizing abstract concepts like vector addition and its resulting magnitude limitations.