Questions: Orthogonal Vectors and Orthonormal Bases

Question 1

You want to find the coordinates of vector v in an orthonormal basis {e₁, e₂, e₃}. What is the correct procedure?

• A. Solve the linear system v = a₁e₁ + a₂e₂ + a₃e₃ for a₁, a₂, a₃ using row reduction
• B. Compute ⟨v, e₁⟩, ⟨v, e₂⟩, ⟨v, e₃⟩ — these inner products directly give the coordinates
• C. Normalize v and then project onto the subspace spanned by each pair of basis vectors
• D. Apply the Gram-Schmidt process to {v, e₁, e₂, e₃} to extract the coordinates

Answer: (B)

With an orthonormal basis, each coordinate is simply ⟨v, eᵢ⟩ — a single inner product computed independently of the others. No system of equations is needed because orthogonality ensures the basis vectors do not 'interfere' with each other. With a non-orthonormal basis (option A), you would need row reduction because the basis vectors are entangled. The orthonormal case eliminates that entanglement entirely.

Question 2

A set of vectors {u₁, u₂} satisfies ⟨u₁, u₂⟩ = 0 and both vectors are nonzero. Is {u₁, u₂} an orthonormal set?

• A. Yes — the zero inner product is the only requirement for orthonormality
• B. Not necessarily — the vectors are orthogonal but may not have unit length
• C. Yes, provided they are also linearly independent
• D. Not necessarily — orthonormal also requires the vectors to span R²

Answer: (B)

Orthonormal requires both conditions: pairwise orthogonality (⟨eᵢ, eⱼ⟩ = 0 for i ≠ j) AND unit length (⟨eᵢ, eᵢ⟩ = 1). The vectors in the question satisfy orthogonality but may have any length. For example, (2, 0) and (0, 3) are orthogonal but not orthonormal. To make them orthonormal, each would need to be divided by its norm.

Question 3

In an orthonormal basis, each coordinate of a vector can be computed independently using a single inner product, without solving a system of equations.

• T. True
• F. False

Answer: True

This is the central computational advantage of orthonormal bases. The formula v = Σ⟨v, eᵢ⟩eᵢ means each component ⟨v, eᵢ⟩ is computable from one dot product. Orthogonality ensures the components do not interact — knowing the 'e₁ component' of v tells you nothing about the 'e₂ component.' With a general basis this independence fails, requiring a linear system to untangle the components.

Question 4

Any set of mutually orthogonal nonzero vectors is automatically an orthonormal set.

• T. True
• F. False

Answer: False

Orthonormal requires unit length in addition to orthogonality. Mutually orthogonal vectors satisfy ⟨uᵢ, uⱼ⟩ = 0 for i ≠ j, but they may have any length. For example, {(2,0,0), (0,3,0), (0,0,5)} is orthogonal but not orthonormal. To obtain an orthonormal set, each vector must be normalized: eᵢ = uᵢ / ‖uᵢ‖.

Question 5

What computational advantage does an orthonormal basis provide over a general basis when finding the coordinates of a vector, and why does orthogonality produce this advantage?

Model answer: With an orthonormal basis, the coordinate corresponding to each basis vector eᵢ is simply the inner product ⟨v, eᵢ⟩ — computable directly, independently of all other coordinates. With a general basis, finding coordinates requires solving a linear system because the basis vectors are not perpendicular and their contributions to v are entangled. Orthogonality eliminates this entanglement: since ⟨eᵢ, eⱼ⟩ = 0 for i ≠ j, projecting onto eᵢ picks up only the eᵢ component of v and nothing from the other directions.

This independence is why orthonormal bases appear throughout numerical linear algebra, signal processing (Fourier series), and quantum mechanics. The coordinate formula v = Σ⟨v, eᵢ⟩eᵢ is simple precisely because orthogonality ensures the basis directions are genuinely independent channels.