Problem 27 - Thomas Calculus

Question

Orthogonal unit vectors If \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) are orthogonal unit vec- tors 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: Understand the Properties of Orthogonal Unit Vectors

Since \(\mathbf{u}_1\) and \(\mathbf{u}_2\) are orthogonal unit vectors, they satisfy two main properties: orthogonality (\(\mathbf{u}_1 \cdot \mathbf{u}_2 = 0\)) and unit norm (\(||\mathbf{u}_1|| = 1\) and \(||\mathbf{u}_2|| = 1\)).

Step 2: Express the Vector \(\mathbf{v}\)

The vector \(\mathbf{v}\) is given as a linear combination of \(\mathbf{u}_1\) and \(\mathbf{u}_2\): \(\mathbf{v} = a \mathbf{u}_1 + b \mathbf{u}_2\).

Step 3: Use Dot Product Properties

To find \(\mathbf{v} \cdot \mathbf{u}_1\), use the distributive property of dot products: \((a \mathbf{u}_1 + b \mathbf{u}_2) \cdot \mathbf{u}_1 = a (\mathbf{u}_1 \cdot \mathbf{u}_1) + b (\mathbf{u}_2 \cdot \mathbf{u}_1)\).

Step 4: Simplify the Dot Product Expression

Since \(\mathbf{u}_1\) and \(\mathbf{u}_2\) are orthogonal unit vectors, \(\mathbf{u}_1 \cdot \mathbf{u}_2 = 0\) and \(\mathbf{u}_1 \cdot \mathbf{u}_1 = 1\). Substitute these into the expression: \(a (1) + b (0) = a\).

Step 5: Conclusion

The value of \(\mathbf{v} \cdot \mathbf{u}_1\) is \(a\). Therefore, the dot product \(\mathbf{v} \cdot \mathbf{u}_1 = a\).

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental mathematical operation that takes two vectors and returns a scalar. It's a measure of how much one vector extends in the direction of another. For two vectors, \( \mathbf{a} \) and \( \mathbf{b} \), the dot product is often written as \( \mathbf{a} \cdot \mathbf{b} \). This operation is calculated using the formula:

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

where \( \theta \) is the angle between the two vectors. This means if the vectors are perpendicular (orthogonal), the dot product will be zero since \( \cos(90^\circ) = 0 \).
This property is crucial in problems involving orthogonal vectors, as it allows us to simplify calculations by taking advantage of the zero product between orthogonal vectors.
In the original exercise, calculating \( \mathbf{v} \cdot \mathbf{u}_1 \) involved using these properties of the dot product to effectively substitute and solve.

Linear Combination

A linear combination in vector mathematics is a composition where vectors are added together after being multiplied by scalars. This concept is used to create new vectors from existing ones in a way that spans across their dimensions. In the given exercise, vector \( \mathbf{v} \) is expressed as a linear combination of orthogonal unit vectors \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \):

• \( \mathbf{v} = a \mathbf{u}_1 + b \mathbf{u}_2 \)

Here, \( a \) and \( b \) are scalars and determine how much of \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) contribute to the vector \( \mathbf{v} \).
This formulation not only preserves the directions of \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) but also leverages their independence because they are orthogonal. As orthogonal vectors provide a unique framework, this ensures each scalar adjusts \( \mathbf{v} \) without interference from the other vector's direction.

Properties of Orthogonal Vectors

Orthogonal vectors have unique characteristics that simplify many calculations in vector mathematics. Two vectors are orthogonal if their dot product is zero. In other words:

• If \( \mathbf{u} \cdot \mathbf{v} = 0 \), then \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal.

Orthogonal vectors are especially useful because they are independent of each other, meaning changes in one vector don't influence the direction of another. This property is crucial in forming bases in vector spaces, where orthogonal bases allow for easier computations.
In the context of unit vectors, orthogonality combined with their unit length (magnitude of 1) provides a powerful tool for resolving components and projections along each vector. In the original exercise, the orthogonality of \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) is key to simplifying the dot product calculation. The orthogonality ensures \( \mathbf{u}_1 \cdot \mathbf{u}_2 = 0 \), allowing us to focus solely on the contribution from \( \mathbf{u}_1 \) to the vector \( \mathbf{v} \). This results in a neat and concise solution.