Problem 29 - University Calculus: Early Transcendentals

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 Orthogonality and Unit Vectors

Orthogonal unit vectors \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) mean that their dot product is zero, i.e., \( \mathbf{u}_1 \cdot \mathbf{u}_2 = 0 \). Each is a unit vector, so \( \| \mathbf{u}_1 \| = 1 \) and \( \| \mathbf{u}_2 \| = 1 \).

Step 2: Express Vector in Basis of Unit Vectors

The vector \( \mathbf{v} = a \mathbf{u}_1 + b \mathbf{u}_2 \) is expressed as a linear combination of \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \), where \( a \) and \( b \) are constants.

Step 3: Compute the Dot Product

The dot product \( \mathbf{v} \cdot \mathbf{u}_1 \) is calculated as follows: \[ \mathbf{v} \cdot \mathbf{u}_1 = (a \mathbf{u}_1 + b \mathbf{u}_2) \cdot \mathbf{u}_1 \]Using the distributive property, \[ = a(\mathbf{u}_1 \cdot \mathbf{u}_1) + b(\mathbf{u}_2 \cdot \mathbf{u}_1) \]Since \( \mathbf{u}_1 \cdot \mathbf{u}_1 = 1 \) and \( \mathbf{u}_2 \cdot \mathbf{u}_1 = 0 \), the expression simplifies to \[ = a \times 1 + b \times 0 = a \]

Step 4: Conclusion

Thus, the value of the dot product is solely determined by the coefficient \( a \), resulting in \( \mathbf{v} \cdot \mathbf{u}_1 = a \).

Key Concepts

Orthogonal Vectors

Orthogonal vectors are fascinating in vector calculus because they meet at right angles. Much like the axes on a graph, they perfectly do not influence each other in terms of direction. Two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal if their dot product equals zero, meaning \( \mathbf{u} \cdot \mathbf{v} = 0 \).
This is a telltale sign of perpendicular vectors.
Why does this matter? If you imagine walking in two different directions that are perfectly orthogonal, your path won’t pull you into the other direction at all.
For example, when you move horizontally, you won’t move vertically unless exerted by another force.
This wonderful property of orthogonal vectors is crucial when expressing other vectors in terms of independent directions.

Dot Product

The dot product is a specific operation between two vectors that results in a scalar quantity.
This scalar gives us the magnitude of projection one vector has on another. It is computed by multiplying corresponding components and summing them.
Mathematically, for vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \), the formula is:
\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]
The dot product has sweet spot applications:

• If the dot product is zero, the vectors are orthogonal.
• It gives insight into the angle between the vectors.
• It is useful in defining vector projections.

The exercise example is a perfect showcase of how dot products reveal when contributions from orthogonal vectors simplify to their scalar components.

Unit Vectors

Unit vectors are the building blocks of vector spaces.
A unit vector has a magnitude of one and directs along a particular direction in space.
It essentially provides direction without extending length.
Such vectors are invaluable for defining coordinates, serving as benchmarks for measuring other vectors.
Technically speaking, if \( \mathbf{u} \) is a unit vector, then \( \| \mathbf{u} \| = 1 \).

In our exercise, \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) are unit vectors.
This property is important because any scaling or reproducibility relies solely on directional modifications rather than magnitude.
Expressing vectors as linear combinations of these unit vectors allows for systematic calculations, as noted in the problem solution.
In essence, they are the compass or bookmarkers for navigation in vector space.