Problem 58 - Precalculus

Question

Let \(\mathbf{u}=\langle a, b\rangle\) be a given vector and suppose that the head of \(\mathbf{n}=\left\langle n_{1}, n_{2}\right\rangle\) lies on the circle \(x^{2}+y^{2}=r^{2} .\) Find the vector \(\mathbf{n}\) such that \(\mathbf{u} \cdot \mathbf{n}\) is as small as possible. Find the actual value of \(\mathbf{u} \cdot \mathbf{n}\) in this case.

Step-by-Step Solution

Answer

The vector is \(-r \frac{\mathbf{u}}{\|\mathbf{u}\|}\) and \( \mathbf{u} \cdot \mathbf{n} = -r\sqrt{a^2 + b^2} \).

Step 1: Understand the Problem

We are given a vector \( \mathbf{u} = \langle a, b \rangle \) and need to find a vector \( \mathbf{n} = \langle n_1, n_2 \rangle \), constrained to lie on the circle \( x^2 + y^2 = r^2 \), such that the dot product \( \mathbf{u} \cdot \mathbf{n} \) is minimized. The circle constraint implies \( n_1^2 + n_2^2 = r^2 \).

Step 2: Formula for Dot Product

The dot product \( \mathbf{u} \cdot \mathbf{n} \) is given by the expression \( a n_1 + b n_2 \). We wish to minimize this expression under the constraint \( n_1^2 + n_2^2 = r^2 \).

Step 3: Use Lagrange Multipliers

To incorporate the constraint \( n_1^2 + n_2^2 = r^2 \), use Lagrange multipliers. Define the Lagrangian \( \mathcal{L} = a n_1 + b n_2 + \lambda (n_1^2 + n_2^2 - r^2) \).

Step 4: Compute Partial Derivatives

Calculate the partial derivatives: \( \frac{\partial \mathcal{L}}{\partial n_1} = a + 2 \lambda n_1 \), \( \frac{\partial \mathcal{L}}{\partial n_2} = b + 2 \lambda n_2 \), \( \frac{\partial \mathcal{L}}{\partial \lambda} = n_1^2 + n_2^2 - r^2 \).

Step 5: Solve System of Equations

Set the partial derivatives to zero: 1) \( a + 2 \lambda n_1 = 0 \), 2) \( b + 2 \lambda n_2 = 0 \), 3) \( n_1^2 + n_2^2 = r^2 \). Solve these equations to find \( n_1 \) and \( n_2 \).

Step 6: Determine \( \lambda \) and Solve for \( n_1, n_2 \)

From equations 1 and 2, express \( n_1 = -\frac{a}{2\lambda} \) and \( n_2 = -\frac{b}{2\lambda} \). Substitute these in the circle equation: \( \left(-\frac{a}{2\lambda}\right)^2 + \left(-\frac{b}{2\lambda}\right)^2 = r^2 \). Solve for \( \lambda \).

Step 7: Calculate Minimum Dot Product

Once the optimal \( n_1 \) and \( n_2 \) are found, calculate the minimum value of \( \mathbf{u} \cdot \mathbf{n} \) using the found \( \mathbf{n} \). The value will be \( \mathbf{u} \cdot \left(-\frac{a}{\sqrt{a^2+b^2}}r, -\frac{b}{\sqrt{a^2+b^2}}r\right) = -r\sqrt{a^2+b^2} \).

Step 8: Final Result

The vector \( \mathbf{n} \) that minimizes \( \mathbf{u} \cdot \mathbf{n} \) is \( \mathbf{n} = -r\frac{\mathbf{u}}{\|\mathbf{u}\|} \). Therefore, the minimized dot product is \( -r\sqrt{a^2 + b^2} \).

Key Concepts

Vector Minimization

Vector minimization involves finding the vector that yields the smallest dot product with a given vector under specific constraints. In the given exercise, we have a vector \( \mathbf{u} = \langle a, b \rangle \) and a vector \( \mathbf{n} = \langle n_1, n_2 \rangle \). The task is to minimize the dot product \( \mathbf{u} \cdot \mathbf{n} \) while keeping \( \mathbf{n} \) on a circle defined by \( x^2 + y^2 = r^2 \).
The dot product \( \mathbf{u} \cdot \mathbf{n} \) calculates as \( an_1 + bn_2 \). A reduced dot product means the vectors are not aligned in the same direction and might point oppositely, which is achieved by manipulation under a constraint. This problem highlights how optimization can be dependent on an additional condition or geometry, like a circle, impacting which vector configuration minimizes the expression.

Lagrange Multipliers

Lagrange multipliers are a crucial tool in vector minimization when constraints are applied. This method helps to locate extrema of functions under specific constraints.
In this exercise, the goal is to minimize the dot product \( \mathbf{u} \cdot \mathbf{n} = an_1 + bn_2 \) with \( \mathbf{n} \) restricted to lie on the circle \( n_1^2 + n_2^2 = r^2 \).
To use Lagrange multipliers, the Lagrangian is defined as \( \mathcal{L} = an_1 + bn_2 + \lambda(n_1^2 + n_2^2 - r^2) \). Here, \( \lambda \) is the multiplier that assists in the balancing of the original function and the constraint. By setting the partial derivatives of \( \mathcal{L} \) with respect to \( n_1, n_2, \) and \( \lambda \) to zero, a system of equations emerges:

• \( a + 2\lambda n_1 = 0 \)
• \( b + 2\lambda n_2 = 0 \)
• \( n_1^2 + n_2^2 = r^2 \)

Solving these equations helps find the vector \( \mathbf{n} \) that minimizes the dot product, demonstrating the power of Lagrange multipliers in robust problem-solving situations involving constraints.

Circle Constraint

A circle constraint implies that the components of the vector \( \mathbf{n} \) must satisfy the equation of a circle, \( n_1^2 + n_2^2 = r^2 \). This geometric restriction ensures the vector aligns on the circle's circumference rather than any arbitrary point.
In vector mathematics, such constraints can limit possibilities for vector formulations but also uniquely determine vector space characteristics. The use of a circle constraint specifically means that solutions to problems, like minimization tasks, must respect the radius \( r \) of the circle - effectively merging geometric and algebraic principles.
In the exercise provided, this means that any possible vector \( \mathbf{n} \) must not only interact with vector \( \mathbf{u} \) but also fulfill a spatial constraint to lie precisely on the circle. This results in a specific combination of vector elements that can minimize \( \mathbf{u} \cdot \mathbf{n} \) and yields a definitive path towards the solution, where the correct choice of vector results in the minimized outcome \( -r\sqrt{a^2 + b^2} \).