Problem 46 - Essential Calculus Early Transcendentals

Question

If \(\mathbf{c}=|\mathbf{a}| \mathbf{b}+|\mathbf{b}| \mathbf{a},\) where \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) are all nonzero vectors, show that \(\mathbf{c}\) bisects the angle between a and b.

Step-by-Step Solution

Answer

Vector \( \mathbf{c} \) bisects the angle between vectors \( \mathbf{a} \) and \( \mathbf{b} \).

Step 1: Understanding the Problem

We are given that \( \mathbf{c} = |\mathbf{a}| \mathbf{b} + |\mathbf{b}| \mathbf{a} \), where \( \mathbf{a}, \mathbf{b}, \text{ and } \mathbf{c} \) are vectors, and we want to show that \( \mathbf{c} \) bisects the angle between \( \mathbf{a} \) and \( \mathbf{b} \). This means that each vector forms equal angles with \( \mathbf{c} \).

Step 2: Expressing the Dot Product and Magnitude

The dot product of two vectors \( \mathbf{v} \) and \( \mathbf{w} \) is \( \mathbf{v} \cdot \mathbf{w} = |\mathbf{v}| |\mathbf{w}| \cos{\theta} \), where \( \theta \) is the angle between them. We also know \( |\mathbf{v}| \) is the magnitude or length of vector \( \mathbf{v} \). We will utilize these to find the angle between \( \mathbf{c} \) and \( \mathbf{a} \) and \( \mathbf{b} \).

Step 3: Calculating the Dot Product of \( \mathbf{c} \) with \( \mathbf{a} \) and \( \mathbf{b} \)

First, find \( \mathbf{c} \cdot \mathbf{a} = (|\mathbf{a}| \mathbf{b} + |\mathbf{b}| \mathbf{a}) \cdot \mathbf{a} = |\mathbf{a}| (\mathbf{b} \cdot \mathbf{a}) + |\mathbf{b}| (\mathbf{a} \cdot \mathbf{a}) \).Similarly, \( \mathbf{c} \cdot \mathbf{b} = (|\mathbf{a}| \mathbf{b} + |\mathbf{b}| \mathbf{a}) \cdot \mathbf{b} = |\mathbf{a}| (\mathbf{b} \cdot \mathbf{b}) + |\mathbf{b}| (\mathbf{a} \cdot \mathbf{b}) \).

Step 4: Simplifying the Expressions

Substitute the magnitudes: \( \mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2 \), \( \mathbf{b} \cdot \mathbf{b} = |\mathbf{b}|^2 \), and note that \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos{\theta} \), where \( \theta \) is the angle between \( \mathbf{a} \) and \( \mathbf{b} \). The expressions become: \[ \mathbf{c} \cdot \mathbf{a} = |\mathbf{a}| (|\mathbf{a}| |\mathbf{b}| \cos{\theta}) + |\mathbf{b}| (|\mathbf{a}|^2) \] \[ \mathbf{c} \cdot \mathbf{b} = |\mathbf{a}| (|\mathbf{b}|^2) + |\mathbf{b}| (|\mathbf{a}| |\mathbf{b}| \cos{\theta}) \].

Step 5: Finding Magnitude of \( \mathbf{c} \)

The magnitude of vector \( \mathbf{c} \) is given by:\[ |\mathbf{c}| = \sqrt{(|\mathbf{a}| \mathbf{b} + |\mathbf{b}| \mathbf{a}) \cdot (|\mathbf{a}| \mathbf{b} + |\mathbf{b}| \mathbf{a})} \].Simplifying, we find:\[ |\mathbf{c}| = \sqrt{ |\mathbf{a}|^2 |\mathbf{b}|^2 + 2 |\mathbf{a}| |\mathbf{b}| (\mathbf{a} \cdot \mathbf{b}) + |\mathbf{b}|^2 |\mathbf{a}|^2 } \] \[ = \sqrt{ (|\mathbf{a}|^2 + |\mathbf{b}|^2) (|\mathbf{b}|^2 + 2 |\mathbf{a}||\mathbf{b}| \cos{\theta} + |\mathbf{a}|^2) } \].

Step 6: Solving for Equal Angles Using Dot Product

The angle \( \phi \) between \( \mathbf{c} \) and \( \mathbf{a} \) is given by \( \cos{\phi} = \frac{\mathbf{c} \cdot \mathbf{a}}{|\mathbf{c}| |\mathbf{a}|} \). Similarly, the angle \( \psi \) between \( \mathbf{c} \) and \( \mathbf{b} \) is \( \cos{\psi} = \frac{\mathbf{c} \cdot \mathbf{b}}{|\mathbf{c}| |\mathbf{b}|} \). After substituting back and comparison:\[ \cos{\phi} = \cos{\psi} \],showing equal angles from \( \mathbf{c} \) bisects the angle between \( \mathbf{a} \) and \( \mathbf{b} \).

Step 7: Conclusion

Since \( \cos{\phi} = \cos{\psi} \), vector \( \mathbf{c} \) bisects the angle between \( \mathbf{a} \) and \( \mathbf{b} \), fulfilling the requirement of angle bisection as both angles from \( \mathbf{a} \) and \( \mathbf{b} \) to \( \mathbf{c} \) are equal.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector algebra that helps us quantify the angle relationship between two vectors. Imagine you have two vectors, \( \mathbf{a} \) and \( \mathbf{b} \), and you want to measure how closely these vectors align with each other. The dot product formula is given by:

• \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos{\theta} \)

where \( |\mathbf{a}| \) and \( |\mathbf{b}| \) are the magnitudes, and \( \theta \) is the angle between them.
This product will be zero if the vectors are perpendicular, positive if they are in a similar direction, and negative if they face away from each other. By using the dot product, you can determine not just the scale of alignment but also expand it to solve real-world problems like angle bisectors.
In the given exercise, calculating the dot product of vector \( \mathbf{c} \) with \( \mathbf{a} \) and \( \mathbf{b} \) was a critical step. This helped to show how \( \mathbf{c} \) relates angle-wise to the other vectors, demonstrating the bisecting property of \( \mathbf{c} \).

Angle Bisection

When a vector bisects the angle between two other vectors, its placement divides the angle into two equal parts. This is not only a geometrical property but also applies to vectors, whereby the bisection implies certain relationships.

• Mathematically, if \( \mathbf{c} \) bisects the angle between \( \mathbf{a} \) and \( \mathbf{b} \), then the angles \( \phi \) and \( \psi \) between \( \mathbf{a} \) and \( \mathbf{c} \) and between \( \mathbf{b} \) and \( \mathbf{c} \), respectively, should be equal.

This concept is crucial in vector calculations as it ensures symmetry.
In the context of the problem, the vector \( \mathbf{c} \) was constructed as a linear combination of \( \mathbf{a} \) and \( \mathbf{b} \), which naturally supported the bisecting property, thanks to the equal magnitude influences from each vector part. This balance is what establishes the bisector role of \( \mathbf{c} \). The equality \( \cos{\phi} = \cos{\psi} \) confirmed the bisecting property by equating the directional influence of each vector on \( \mathbf{c} \).
Understanding angle bisection goes beyond the geometric intuition and requires recognizing how vector operations support these influences.

Magnitude of Vectors

Vectors have magnitudes, which represent their length or the extent from one point to another in space. The magnitude is a scalar and simplifies understanding how value scales in dimensional analysis. It's computed as:

• \( |\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + ... + a_n^2} \)

for a vector \( \mathbf{a} = (a_1, a_2, ..., a_n) \). Magnitude is crucial in determining the span or strength of a vector.
In solving the exercise, we used the concept to define the construction of \( \mathbf{c} \) which weighted vector \( \mathbf{a} \) by the magnitude of \( \mathbf{b} \) and vice-versa. Such operations ensure vector \( \mathbf{c} \) is proportionally influenced, making magnitude essential for ensuring the balance needed for angle bisection.
By confirming \( \mathbf{c} \) as a sum of scaled versions of \( \mathbf{a} \) and \( \mathbf{b} \), the magnitude played a pivotal role in solidifying \( \mathbf{c} \)'s properties to adequately prove its bisecting role. Understanding how magnitude functions helps in visualizing the solid form of vectors during such tasks.