Problem 26 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Define the Known Variables

Given vectors \( \mathbf{u} \) and \( \mathbf{v} \) such that \( |\mathbf{u}| = |\mathbf{v}| \). We know these vectors form a parallelogram, and we need to determine if a diagonal of this parallelogram bisects the angle between the vectors.

Step 2: Construct the Diagonal Expression

The diagonal of the parallelogram formed by vectors \(\mathbf{u}\) and \(\mathbf{v}\) is given by \( \mathbf{d} = \mathbf{u} + \mathbf{v} \). This vector represents one of the diagonals of the parallelogram.

Step 3: Condition for Angle Bisector

The vector \( \mathbf{d} \) bisects the angle between \( \mathbf{u} \) and \( \mathbf{v} \) if the angle formed between \( \mathbf{d} \) and \( \mathbf{u} \) is equal to the angle formed between \( \mathbf{d} \) and \( \mathbf{v} \). This holds if the dot products between these vectors satisfy the proportion: \( \mathbf{u} \cdot \mathbf{d} = \mathbf{v} \cdot \mathbf{d} \).

Step 4: Calculate Dot Products

Calculate the dot products: \( \mathbf{u} \cdot \mathbf{d} = \mathbf{u} \cdot (\mathbf{u} + \mathbf{v}) = \mathbf{u} \cdot \mathbf{u} + \mathbf{u} \cdot \mathbf{v} \),\( \mathbf{v} \cdot \mathbf{d} = \mathbf{v} \cdot (\mathbf{u} + \mathbf{v}) = \mathbf{v} \cdot \mathbf{u} + \mathbf{v} \cdot \mathbf{v} \).

Step 5: Apply the Condition and Simplify

Since \( |\mathbf{u}| = |\mathbf{v}| \), it follows that \( \mathbf{u} \cdot \mathbf{u} = \mathbf{v} \cdot \mathbf{v} \). Substituting, the expressions for dot products become equal: \( \mathbf{u} \cdot \mathbf{u} + \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} + \mathbf{v} \cdot \mathbf{v} \). Since \( \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \), this equality is satisfied.

Step 6: Conclusion

Thus, by showing that \( \mathbf{u} \cdot \mathbf{d} = \mathbf{v} \cdot \mathbf{d} \) and noting \( |\mathbf{u}| = |\mathbf{v}| \), we have proven that \( \mathbf{d} = \mathbf{u} + \mathbf{v} \) bisects the angle between \( \mathbf{u} \) and \( \mathbf{v} \).

Key Concepts

Parallelogram

A parallelogram is a four-sided shape with opposite sides that are both equal in length and parallel.
In vector geometry, a parallelogram can be created using two vectors, typically denoted as \( \mathbf{u} \) and \( \mathbf{v} \).
These vectors serve as sides of the parallelogram. The other pair of sides is formed by translating these vectors parallelly through space. Key properties of a parallelogram include:

• Opposite angles are equal.
• The diagonals bisect each other.
• Area can be found using the cross product of the forming vectors.

Understanding how vectors form a parallelogram is vital in vector addition and many geometry problems.

Vector Addition

Vector addition is a fundamental operation in vector geometry.
It involves combining two or more vectors to find a resultant vector. The resultant vector is also known as the diagonal in the context of a parallelogram formed by the vectors. To perform vector addition, you simply add the corresponding components of the vectors. For vectors \( \mathbf{u} = (u_1, u_2) \) and \( \mathbf{v} = (v_1, v_2) \), the sum \( \mathbf{d} = \mathbf{u} + \mathbf{v} \) is calculated as:

• \( d_1 = u_1 + v_1 \)
• \( d_2 = u_2 + v_2 \)

This addition gives a new vector, \( \mathbf{d} \) that represents one diagonal of the parallelogram. This principle is particularly useful in physics and engineering where vectors describe everything from forces to velocities.

Dot Product

The dot product, also known as the scalar product, is an operation that takes two vectors and returns a scalar.
It is computed as the sum of the products of the corresponding entries of the vectors. For vectors \( \mathbf{u} = (u_1, u_2) \) and \( \mathbf{v} = (v_1, v_2) \), the dot product is calculated as:

• \( \mathbf{u} \cdot \mathbf{v} = u_1 \times v_1 + u_2 \times v_2 \)

The dot product is useful in finding the angle between two vectors, as it relates to the cosine of that angle. If the dot product is zero, the vectors are perpendicular. When the task asks to check if the diagonal bisects the angle between two vectors, calculating the dot products helps confirm this property by comparing values.

Angle Bisector

An angle bisector is a line or vector that divides an angle into two equal angles.
In the context of a parallelogram, the diagonal can act as an angle bisector if it evenly splits the angle formed by the two adjacent vectors. To verify if a vector, such as the diagonal \( \mathbf{d} = \mathbf{u} + \mathbf{v} \), bisects the angle between vectors \( \mathbf{u} \) and \( \mathbf{v} \), one must show the dot products satisfy:

• \( \mathbf{u} \cdot \mathbf{d} = \mathbf{v} \cdot \mathbf{d} \)

This condition arises from geometric properties and assures the diagonal slices the angle perfectly in two equal sections.
Recognizing a diagonal as an angle bisector can simplify solving geometric problems involving symmetry and equality of angles.