Problem 31 - Thomas Calculus

Question

Orthogonal unit vectors If \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) are orthogonal unit vectors and \(\mathbf{v}=a \mathbf{u}_{1}+b \mathbf{u}_{2},\) find \(\mathbf{v} \cdot \mathbf{u}_{1} .\)

Step-by-Step Solution

Answer

\( \mathbf{v} \cdot \mathbf{u}_{1} = a \).

Step 1: Understanding Orthogonal Vectors

Orthogonal vectors are vectors that are perpendicular to each other. For vectors \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) to be orthogonal, their dot product must be zero, i.e., \(\mathbf{u}_{1} \cdot \mathbf{u}_{2} = 0\).

Step 2: Identifying Unit Vectors

Unit vectors have a magnitude of 1. This means that \(\|\mathbf{u}_{1}\| = 1\) and \(\|\mathbf{u}_{2}\| = 1\).

Step 3: Express Vector \(\mathbf{v}\)

The vector \(\mathbf{v}\) is given by \(\mathbf{v} = a \mathbf{u}_{1} + b \mathbf{u}_{2}\), where \(a\) and \(b\) are scalars.

Step 4: Calculate \(\mathbf{v} \cdot \mathbf{u}_{1}\)

To find \(\mathbf{v} \cdot \mathbf{u}_{1}\), use the distributive property of the dot product: \(\mathbf{v} \cdot \mathbf{u}_{1} = (a \mathbf{u}_{1} + b \mathbf{u}_{2}) \cdot \mathbf{u}_{1}\).

Step 5: Simplify Using Dot Product Properties

Distribute the dot product: \((a \mathbf{u}_{1} + b \mathbf{u}_{2}) \cdot \mathbf{u}_{1} = a(\mathbf{u}_{1} \cdot \mathbf{u}_{1}) + b(\mathbf{u}_{2} \cdot \mathbf{u}_{1})\).

Step 6: Apply Orthogonal and Unit Vector Properties

Since \(\mathbf{u}_{1} \cdot \mathbf{u}_{1} = 1\) (unit vector property), and \(\mathbf{u}_{2} \cdot \mathbf{u}_{1} = 0\) (orthogonality), the expression simplifies to \(a \cdot 1 + b \cdot 0 = a\).

Key Concepts

Unit Vectors

Unit vectors are fundamental in vector mathematics. They are vectors that have a magnitude, or length, of exactly 1. This characteristic makes them very useful in defining directions in space without scaling. In a Cartesian coordinate system, the unit vectors along the x, y, and z axes are often denoted as \(\mathbf{i}, \mathbf{j},\ and \, \mathbf{k}\) respectively, with each representing a different axis.

A unit vector \(\mathbf{u}\) can be found by dividing a vector \(\mathbf{v}\) by its magnitude, which is represented mathematically as \(\mathbf{u} = \frac{\mathbf{v}}{\| \mathbf{v} \|}\). This operation results in a vector that matches the direction of \(\mathbf{v}\) but with a magnitude of 1.

• important for defining directions
• used to create orthogonal unit vectors
• helpful in vector projections

Dot Product

The dot product, also known as the scalar product, is a way of multiplying two vectors that results in a scalar (a single number). Calculated by \(\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta\), where \(\theta\) is the angle between the vectors. If \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, \(\theta = 90^\circ\), and so \(\cos \theta = 0\), meaning their dot product is zero.

The properties of the dot product make it particularly useful for identifying orthogonal vectors. For example, when computing \(\mathbf{v} \cdot \mathbf{u}_1\), we apply the distributive property to evaluate contributions from each component.

• produces a scalar
• useful in projections and finding angles
• zero result indicates orthogonality

Magnitude

The magnitude of a vector represents its length and is crucial in many vector operations. For a vector \( \mathbf{v} = (x, y, z) \), the magnitude is found using the formula \(\| \mathbf{v} \| = \sqrt{x^2 + y^2 + z^2}\). This formula derives from the Pythagorean theorem and works in any dimensional space.

The magnitude is important because it allows us to normalize vectors to unit length, simplifying many mathematical processes, like creating unit vectors.

• measures vector length
• helps in normalizing vectors
• essential for accurate distance computation

Linear Combination

A linear combination involves using scalar multiplication and vector addition to construct a new vector. For instance, given vectors \(\mathbf{u}_1\) and \(\mathbf{u}_2\), and scalars \(a\) and \(b\), a linear combination \(\mathbf{v}\) is represented as \(\mathbf{v} = a \mathbf{u}_1 + b \mathbf{u}_2\). This principle underlies much of vector and mathematical theory, as it allows for the mixing of different directions and magnitudes.

Linear combinations are not only central in forming new vectors but are also key in understanding vector spaces and transformations. By analyzing \(\mathbf{v}\), one can understand the impact of each individual component on the resultant vector.

• constructs vectors from others
• useful in systems of equations and transformations
• explores capabilities of vector spaces