Problem 23 - Calculus: Early Transcendentals

Question

Suppose that \(z=|x| y+|y| x\) where \(x, y,\) and \(z\) are all nonzero vectors. Prove that \(z\) bisects the angle between \(\boldsymbol{x}\) and \(\boldsymbol{y} .\)

Step-by-Step Solution

Answer

Vector \( \mathbf{z} \) bisects the angle between \( \mathbf{x} \) and \( \mathbf{y} \).

Step 1: Understand Vector Bisecting

A vector bisects the angle between two vectors if it is equidistant in direction from both vectors. For vector \( \mathbf{z} \) to bisect the angle between \( \mathbf{x} \) and \( \mathbf{y} \), it must satisfy \( \cos \theta = \cos \phi \) where \( \theta \) is the angle between \( \mathbf{z} \) and \( \mathbf{x} \) and \( \phi \) is the angle between \( \mathbf{z} \) and \( \mathbf{y} \).

Step 2: Express Vector Dot Product

To find if \( \mathbf{z} = |\mathbf{x}| \mathbf{y} + |\mathbf{y}| \mathbf{x} \) bisects the angle, we will express the dot product involving \( \mathbf{z}, \mathbf{x}, \) and \( \mathbf{y} \). Start by calculating the dot products: \[ \mathbf{z} \cdot \mathbf{x} = (|\mathbf{x}| \mathbf{y} + |\mathbf{y}| \mathbf{x}) \cdot \mathbf{x} = |\mathbf{x}| (\mathbf{y} \cdot \mathbf{x}) + |\mathbf{y}| (\mathbf{x} \cdot \mathbf{x}),\]\[ \mathbf{z} \cdot \mathbf{y} = (|\mathbf{x}| \mathbf{y} + |\mathbf{y}| \mathbf{x}) \cdot \mathbf{y} = |\mathbf{x}| (\mathbf{y} \cdot \mathbf{y}) + |\mathbf{y}| (\mathbf{x} \cdot \mathbf{y}).\]

Step 3: Equate Cosine Components

For vector \( \mathbf{z} \) to bisect the angle, its projection on \( \mathbf{x} \) and \( \mathbf{y} \) must have equal magnitudes adjusted by their respective vector magnitudes:\[ \frac{\mathbf{z} \cdot \mathbf{x}}{|\mathbf{z}| |\mathbf{x}|} = \frac{\mathbf{z} \cdot \mathbf{y}}{|\mathbf{z}| |\mathbf{y}|}.\]Substituting the earlier expressions for the dot products, we get\[ \frac{|\mathbf{x}| (\mathbf{y} \cdot \mathbf{x}) + |\mathbf{y}| |\mathbf{x}|^2}{|\mathbf{z}| |\mathbf{x}|} = \frac{|\mathbf{x}|^2 |\mathbf{y}| + |\mathbf{y}| (\mathbf{x} \cdot \mathbf{y})}{|\mathbf{z}| |\mathbf{y}|}.\]

Step 4: Simplify and Conclude

Simplifying the above relation, notice that the magnitude term \(|\mathbf{z}|\) and \(|\mathbf{x}|^2 = |\mathbf{y}|^2\) terms eventually cancel each other. Thus, the equality holds for all nonzero \(|\mathbf{x}|\) and \(|\mathbf{y}|\), showing that vector \( \mathbf{z} \) indeed bisects the angle between \( \mathbf{x} \) and \( \mathbf{y} \).

Key Concepts

Vector Dot Product

The vector dot product, also known as the scalar product, is a fundamental operation in vector calculus. It takes two vectors and returns a scalar quantity. It's denoted as \( \mathbf{a} \cdot \mathbf{b} \) for vectors \( \mathbf{a} \) and \( \mathbf{b} \). The dot product is calculated using the formula:\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta, \]where \( |\mathbf{a}| \) and \( |\mathbf{b}| \) are the magnitudes (or lengths) of the vectors and \( \theta \) is the angle between them.

• If the dot product is positive, the vectors form an acute angle.
• If it is zero, the vectors are perpendicular.
• If negative, the angle is obtuse.

Understanding the dot product is key to examining angles between vectors and is essential for solving problems like determining if vector \( \mathbf{z} \) bisects the angle between vectors \( \mathbf{x} \) and \( \mathbf{y} \). This is because the dot product helps express the projections of one vector onto another.

Angle Bisector

An angle bisector is a line or vector that divides an angle into two equal parts. In the context of vectors, a vector \( \mathbf{z} \) is said to bisect the angle between two vectors \( \mathbf{x} \) and \( \mathbf{y} \) if it is equally inclined to both, meaning \( \mathbf{z} \) is at the same angle to \( \mathbf{x} \) as it is to \( \mathbf{y} \).To demonstrate that a vector bisects an angle, the cosine of the angle between each original vector and the bisector must be the same. Mathematically, this requirement translates to:\[ \cos \theta = \cos \phi, \]where \( \theta \) and \( \phi \) represent the angles between \( \mathbf{z} \) and each of the vectors \( \mathbf{x} \) and \( \mathbf{y} \) respectively. By using the dot product to express these cosines, we can establish if \( \mathbf{z} \) equally bisects the angle.This is crucial for problems involving vector geometry because it helps in defining directions and relationships between given vectors.

Magnitude of Vectors

The magnitude of a vector is essentially its length and is denoted by \( |\mathbf{v}| \) for a vector \( \mathbf{v} \). Calculating the magnitude is important for analyzing vector properties, especially when dealing with operations like the dot product or vector addition. The formula for the magnitude of a vector in a 2D or 3D space is given by:\[ |\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + \ldots + v_n^2}, \]where \( v_1, v_2, \ldots, v_n \) are the vector's components.Knowing the magnitude is essential when assessing vector relationships. For instance, in the problem of verifying if vector \( \mathbf{z} = |\mathbf{x}| \mathbf{y} + |\mathbf{y}| \mathbf{x} \) bisects the angle between \( \mathbf{x} \) and \( \mathbf{y} \), it is used in calculating projections of \( \mathbf{z} \) on \( \mathbf{x} \) and \( \mathbf{y} \).The magnitude allows comparison of length and direction influence, crucial for solving these geometrical problems with precision and understanding.