Question

Assume \(\mathbf{u}\) and \(\mathbf{v}\) are nonzero vectors that are not parallel. Show that \(\mathbf{w}=\|\mathbf{u}\| \mathbf{v}+\|\mathbf{v}\| \mathbf{u}\) is a nonzero vector that bisects the angle between \(\mathbf{u}\) and \(\mathbf{v}\).

Short answer

Vector \(\mathbf{w}\) is nonzero and bisects the angle between \(\mathbf{u}\) and \(\mathbf{v}\).

Step-by-step solution

Find the Magnitudes

Calculate the magnitudes of vectors \(\mathbf{u}\) and \(\mathbf{v}\), which are \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\) respectively.

Analyze the Direction of \(\mathbf{w}\)

\(\mathbf{w} = \|\mathbf{u}\| \mathbf{v} + \|\mathbf{v}\| \mathbf{u}\). This expression shows that \(\mathbf{w}\) is a linear combination of \(\mathbf{u}\) and \(\mathbf{v}\). Both \(\|\mathbf{u}\| \mathbf{v}\) and \(\|\mathbf{v}\| \mathbf{u}\) are vectors pointing in the direction of \(\mathbf{v}\) and \(\mathbf{u}\) respectively, each scaled by the other's magnitude.

Prove \(\mathbf{w}\) is Non-zero

Since \(\mathbf{u}\) and \(\mathbf{v}\) are nonzero vectors, and they are not parallel, the vectors \(\|\mathbf{u}\| \mathbf{v}\) and \(\|\mathbf{v}\| \mathbf{u}\) are also non-zero and non-parallel. Therefore, their sum \(\mathbf{w}\) is non-zero.

Find the Bisector Condition

A vector \(\mathbf{w}\) bisects the angle between \(\mathbf{u}\) and \(\mathbf{v}\) if it satisfies \(\mathbf{w} = a \mathbf{u} + b \mathbf{v}\) for some positive constants \(a\) and \(b\), such that \(\frac{a}{b} = \frac{\|\mathbf{v}\|}{\|\mathbf{u}\|}\).

Verify the Bisector Condition for \(\mathbf{w}\)

Given \(\mathbf{w} = \|\mathbf{u}\| \mathbf{v} + \|\mathbf{v}\| \mathbf{u}\), compare it to \(a\mathbf{u} + b\mathbf{v}\), where \(a = \|\mathbf{v}\|\) and \(b = \|\mathbf{u}\|\). The ratio \(\frac{a}{b} = \frac{\|\mathbf{v}\|}{\|\mathbf{u}\|}\), satisfying our condition, hence \(\mathbf{w}\) is the angle bisector.

Key concepts

Non-parallel Vectors

Understanding non-parallel vectors is crucial when dealing with vector operations like addition or finding bisectors. Vectors are non-parallel when they do not lie on the same line or direction. This means that one vector cannot be expressed as a scalar multiple of the other.

Non-parallel vectors are important because they span a plane, creating the potential for interesting geometric interpretations such as finding intersections, angles, and bisectors. When working with vectors like \( \mathbf{u} \) and \( \mathbf{v} \), knowing they're non-parallel confirms they can form an angle, which is vital for understanding concepts such as direction and magnitude.

In this exercise, the non-parallel nature of \( \mathbf{u} \) and \( \mathbf{v} \) ensures that the vector \( \mathbf{w} = \| \mathbf{u} \| \mathbf{v} + \| \mathbf{v} \| \mathbf{u} \) can indeed bisect the angle between them. This is because they have distinct directional components, allowing \( \mathbf{w} \) to find the equal angle between them.

Linear Combination

A linear combination involves creating a new vector by scaling and adding vectors together. This is represented as \( a \mathbf{u} + b \mathbf{v} \), where \( a \) and \( b \) are scalars. In our exercise, the vector \( \mathbf{w} = \| \mathbf{u} \| \mathbf{v} + \| \mathbf{v} \| \mathbf{u} \) is formed as a linear combination of vectors \( \mathbf{u} \) and \( \mathbf{v} \).

The scalars \( a = \| \mathbf{v} \| \) and \( b = \| \mathbf{u} \| \) are magnitudes of the respective vectors, ensuring that \( \mathbf{w} \) maintains a balance of both input vectors.

This technique of forming linear combinations is pivotal in vector spaces as it helps in expressing any vector through the sum of others. Such combinations enable transformations and provide solutions like determining the bisector vector \( \mathbf{w} \) here.

Magnitude of Vectors

The magnitude of a vector, often represented as \( \| \mathbf{u} \| \) for vector \( \mathbf{u} \), measures its length or size. It's computed using the formula \( \sqrt{x^2 + y^2 + z^2} \) for a vector in three-dimensional space. For two-dimensional vectors, the formula becomes \( \sqrt{x^2 + y^2} \).

Understanding magnitude is essential because it scales vectors, allowing them to represent quantities accurately in problems like force, velocity, or in this exercise, the vector \( \mathbf{w} \).

• Magnitude provides the information on the "strength" of a vector.
• It allows comparisons between vector sizes.
• It aids in operations such as finding angle bisectors.

In our example, the magnitudes \( \| \mathbf{u} \| \) and \( \| \mathbf{v} \| \) scale the vectors \( \mathbf{v} \) and \( \mathbf{u} \), respectively, ensuring that their contributions to \( \mathbf{w} \) are balanced. This balance, characterized by maintaining proportional magnitudes, is crucial for \( \mathbf{w} \) to bisect the angle properly between \( \mathbf{u} \) and \( \mathbf{v} \).

Thus, the understanding of vector magnitude allows us to solidify why \( \mathbf{w} \) is not only formed but can function as a bisector due to these respected vector lengths.