Problem 22 - Thomas Calculus

Question

Diagonal of parallelogram Show that the indicated diagonal of the parallelogram determined by vectors \(u\) and \(v\) bisects the angle between \(\mathbf{u}\) and \(\mathbf{v}\) if \(|\mathbf{u}|=|\mathbf{v}|\)

Step-by-Step Solution

Answer

The diagonal \( \mathbf{u} + \mathbf{v} \) bisects the angle if \(|\mathbf{u}| = |\mathbf{v}|\).

Step 1: Identify Diagonal and Condition

In a parallelogram with vectors \( \mathbf{u} \) and \( \mathbf{v} \), the diagonals can be expressed as \( \mathbf{u} + \mathbf{v} \) and \( \mathbf{u} - \mathbf{v} \). We need to show that the diagonal \( \mathbf{u} + \mathbf{v} \) bisects the angle between \( \mathbf{u} \) and \( \mathbf{v} \) given the condition \( |\mathbf{u}| = |\mathbf{v}| \).

Step 2: Equilibrium of Angles for Bisecting

For a vector to bisect the angle between two other vectors, the angle subtended by the two vectors to the bisector should be equal. Therefore, we need to prove that \( \mathbf{u} \cdot (\mathbf{u} + \mathbf{v}) = \mathbf{v} \cdot (\mathbf{u} + \mathbf{v}) \).

Step 3: Set Up Dot Product Equations

Starting with the dot products, expand both: \( \mathbf{u} \cdot (\mathbf{u} + \mathbf{v}) = \mathbf{u} \cdot \mathbf{u} + \mathbf{u} \cdot \mathbf{v} \) and \( \mathbf{v} \cdot (\mathbf{u} + \mathbf{v}) = \mathbf{v} \cdot \mathbf{u} + \mathbf{v} \cdot \mathbf{v} \).

Step 4: Use Given Magnitude Condition

Since \( |\mathbf{u}| = |\mathbf{v}| \), we know \( \mathbf{u} \cdot \mathbf{u} = \mathbf{v} \cdot \mathbf{v} \). Substitute these values into the previous equations: \( \mathbf{u} \cdot \mathbf{u} + \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} + \mathbf{v} \cdot \mathbf{v} \).

Step 5: Simplify and Conclude

The expression simplifies to \( \mathbf{u} \cdot \mathbf{u} = \mathbf{v} \cdot \mathbf{v} \), as the cross terms \( \mathbf{u} \cdot \mathbf{v} \) cancel each other out. This is true by the given condition \( |\mathbf{u}| = |\mathbf{v}| \), thus proving that the diagonal \( \mathbf{u} + \mathbf{v} \) bisects the angle between \( \mathbf{u} \) and \( \mathbf{v} \).

Key Concepts

Dot Product

The dot product is a fundamental concept in vector calculus. It is a scalar representation of the product of two vectors. The dot product of vectors \( \mathbf{a} \) and \( \mathbf{b} \) is calculated using the formula: \[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta) \] Here, \( |\mathbf{a}| \) and \( |\mathbf{b}| \) are the magnitudes of the vectors, and \( \theta \) is the angle between them.

• The dot product is commutative, meaning \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \).
• It is zero if the vectors are perpendicular.
• You can find it using components of vectors: \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \), giving \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \).

In the given exercise, you use dot product properties to show the diagonal of a parallelogram bisects the angle between two vectors if their magnitudes are equal.
It shows balance when resolving into components, leading to simplified equalities as seen in the steps.

Parallelogram Law

The parallelogram law is a geometric rule that helps in understanding how vectors add. When you place two vectors, \( \mathbf{u} \) and \( \mathbf{v} \), tail to tail, their sum forms the diagonal of the parallelogram. The two diagonals are given by:

• \( \mathbf{u} + \mathbf{v} \)
• \( \mathbf{u} - \mathbf{v} \)

This visual aids in finding resultant vectors and understanding vector relationships in both magnitude and direction.

• It shows symmetry in vector addition.
• The condition \( |\mathbf{u}| = |\mathbf{v}| \) simplifies results in tasks like bisecting angles.

Applying this law, the exercise explains how the diagonal \( \mathbf{u} + \mathbf{v} \) perfectly bisects the angle when magnitudes are equal.
This balanced addition shows that each side of the parallelogram contributes equally to the diagonal, highlighting the parallel relationships in vectors.

Vector Magnitude

Understanding vector magnitude is crucial in vector calculations. The magnitude or length of a vector \( \mathbf{u} = (u_1, u_2) \) is determined by the formula: \[ |\mathbf{u}| = \sqrt{u_1^2 + u_2^2} \] This measure helps in comparing vectors, calculating the dot product, and analyzing vector components.

• Magnitude is always a positive quantity or zero.
• It represents the "size" or "length" of the vector in space.
• Equal magnitudes imply that two vectors have the same length irrespective of their direction.

In the exercise's context, the equality \( |\mathbf{u}| = |\mathbf{v}| \) was pivotal.
This fact was used to conclude that the angles subtended by the diagonals to the vectors were equal, proving they bisect the specified angle.
Understanding these properties is key to mastering vector problems, where directionality and size play a significant role.