Problem 21 - APEX Calculus

Question

Under what conditions is \(\|\vec{u}\|+\|\vec{v}\|=\|\vec{u}+\vec{v}\| ?\)

Step-by-Step Solution

Answer

The equation holds if \(\vec{u}\) and \(\vec{v}\) are parallel vectors (collinear).

Step 1: Understanding the Norm Property

The expression \(\|\vec{u}\|+\|\vec{v}\|=\|\vec{u}+\vec{v}\|\) relates to the triangle inequality in vector spaces. The triangle inequality states that \(\|\vec{u} + \vec{v}\| \leq \|\vec{u}\| + \|\vec{v}\|\). The given equation asks under what conditions this inequality becomes an equality.

Step 2: Condition for Equality in Triangle Inequality

The triangle inequality becomes an equality if and only if the vectors \(\vec{u}\) and \(\vec{v}\) are collinear, meaning one is a scalar multiple of the other. Mathematically, this can be expressed as \( \vec{v} = k\vec{u} \) or \( \vec{u} = k\vec{v} \), where \(k\) is a scalar.

Step 3: Conclusion

Thus, the expression \(\|\vec{u}\|+\|\vec{v}\|=\|\vec{u} + \vec{v}\|\) holds true if and only if \(\vec{u}\) and \(\vec{v}\) are parallel vectors. This means there exists a scalar \(k\) such that \(\vec{v} = k\vec{u}\) or \(\vec{u} = k\vec{v}\).

Key Concepts

Triangle Inequality

The triangle inequality is a fundamental concept in vector spaces. It states that for any vectors \( \vec{u} \) and \( \vec{v} \), the magnitude of their sum is always less than or equal to the sum of their magnitudes: \( \|\vec{u} + \vec{v}\| \leq \|\vec{u}\| + \|\vec{v}\| \). This inequality resembles the principle of a triangle, where the length of one side is never greater than the sum of the other two sides.
In the context of our exercise, the expression \( \|\vec{u}\| + \|\vec{v}\| = \|\vec{u} + \vec{v}\| \) asks us to investigate when this inequality becomes an equality. It translates to situations where you can "walk" from the tip of one vector to the tip of the other by walking along their combined path directly.
Understanding when this is possible helps unveil the special structural relationships between vectors, essentially determining when vectors are acting along the same line.

Collinear Vectors

Collinear vectors are vectors that lie on the same line, which implies they are scalar multiples of one another. This concept is vital for understanding the conditions that make the triangle inequality become an equality. If vectors \( \vec{u} \) and \( \vec{v} \) are collinear, there exists a scalar \( k \) such that \( \vec{v} = k\vec{u} \) or \( \vec{u} = k\vec{v} \). This means if you multiply one vector by some scalar, you will obtain the other.
When vectors are collinear:

• They either point in the same or opposite directions.
• They perfectly "add up," not bending or forming an angle between them.
• The triangle inequality becomes an equality: \( \|\vec{u}\| + \|\vec{v}\| = \|\vec{u} + \vec{v}\| \).

Checking for collinearity can be as simple as confirming one vector is a scalar multiple of the other, making it a straightforward task to identify these special scenarios in vector algebra.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar, which is simply a real number. This operation scales the vector by stretching or shrinking it, and also has the possibility of reversing its direction if the scalar is negative.
It's denoted mathematically as \( k\vec{v} \), where \( k \) is a scalar and \( \vec{v} \) is a vector. Important aspects of scalar multiplication include:

• The direction of the vector is unchanged when \( k > 0 \), but reversed when \( k < 0 \).
• The magnitude of the vector becomes \( |k| \times \|\vec{v}\| \).
• If \( k = 1 \), the vector remains unchanged, while \( k = 0 \) results in the zero vector.

In our exercise context, scalar multiplication's role explains the occurrence of collinear vectors. By multiplying one vector by a suitable scalar, the vectors become aligned, allowing the triangle inequality to transform into an equality. This simple yet powerful operation showcases how vectors interrelate under certain conditions, fundamentally connecting scalar multiplication with vector collinearity.