Problem 98 - Precalculus

Question

Show that if \(\mathbf{u}=a \mathbf{i}+b \mathbf{j}\) is a unit vector, then \((a, b)\) lies on the unit circle.

Step-by-Step Solution

Answer

If \( \mathbf{u} \) is a unit vector, then \((a, b)\) satisfies \(a^2 + b^2 = 1\), proving it lies on the unit circle.

Step 1: Understanding a Unit Vector

A unit vector is a vector with a magnitude of 1. If \( \mathbf{u} = a \mathbf{i} + b \mathbf{j} \) is a unit vector, then the length (or magnitude) of \( \mathbf{u} \) must be 1. This means \( \| \mathbf{u} \| = 1 \).

Step 2: Calculating the Magnitude

The magnitude of vector \( \mathbf{u} = a \mathbf{i} + b \mathbf{j} \) is given by \( \| \mathbf{u} \| = \sqrt{a^2 + b^2} \). Since \( \mathbf{u} \) is a unit vector, this magnitude must equal 1: \( \sqrt{a^2 + b^2} = 1 \).

Step 3: Squaring Both Sides

To eliminate the square root, we square both sides of the equation: \((\sqrt{a^2 + b^2})^2 = 1^2 \). This simplifies to \( a^2 + b^2 = 1 \).

Step 4: Conclusion

The equation \( a^2 + b^2 = 1 \) represents the equation of a unit circle centered at the origin (0,0) in the xy-plane. Therefore, \((a, b)\) lies on the unit circle.

Key Concepts

Understanding a Unit Vector

A unit vector is a fundamental concept in vector mathematics that simplifies complex systems by providing a vector with a magnitude of exactly 1. This makes unit vectors extremely useful for representing direction without scaling effects. For example, if you have a vector \( \mathbf{u} = a \mathbf{i} + b \mathbf{j} \), and it's stated that this vector is a unit vector, this directly implies that the vector's

• length or magnitude is precisely 1.
• direction is maintained without affecting its size.

Unit vectors serve multiple purposes, especially where direction is needed in coordinate systems or when normalizing other vectors. By ensuring the magnitude is 1, calculations become straightforward and universally applicable across different vector magnitudes. An application could be when determining directional angles and positions using trigonometric functions.

Magnitude of a Vector

The magnitude of a vector gives you a measure of its length in a coordinate system. In geometry and physics, the magnitude is important since it often represents the size of a force, speed, or distance. For the vector \( \mathbf{u} = a \mathbf{i} + b \mathbf{j} \), you find the magnitude using the formula:\[ \| \mathbf{u} \| = \sqrt{a^2 + b^2} \].For unit vectors, this magnitude must equate to 1, hence simplifying our original problem drastically, since it implies the equation:\[ \sqrt{a^2 + b^2} = 1 \].By squaring both sides, you get rid of the square root, resulting in:\[ a^2 + b^2 = 1 \].With this equation, any vector that satisfies it is guaranteed to be a unit vector. Understanding the magnitude also aids in discerning how vectors relate in space, whether they are closer or further apart.

Coordinate Geometry Essentials

Coordinate geometry, sometimes known as analytic geometry, combines algebraic techniques and principles with geometrical figures. This branch of geometry focuses on describing geometric figures in a coordinate plane and can involve the use of the equation of shapes, distances, and angles. When examining a vector such as \( \mathbf{u} = a \mathbf{i} + b \mathbf{j} \), coordinate geometry allows us to place this vector in the XY-plane. The point \( (a, b) \) is then a position on this plane derived from its components.Coordinate geometry is significant as it translates visual shapes into mathematical equations and functions. From representing a circle or an ellipse to finding intersections and distances, this field provides a powerful technique to solve spatial problems in a structured way.

Equation of a Circle

The equation of a circle is central to understanding geometric shapes’ constraints in a coordinate system. The standard form of a circle's equation in the plane is \( (x - h)^2 + (y - k)^2 = r^2 \).

• The center is \( (h, k) \).
• The radius is \( r \).

For a unit circle, centered at the origin with radius 1 (which is common in unit vector problems), this simplifies to:\[ x^2 + y^2 = 1 \].This form shows that for any point on this circle, the sum of the squares of the coordinates equals the square of the radius—here, 1. Applying this to the vector \( \mathbf{u} = a \mathbf{i} + b \mathbf{j} \), if \( (a, b) \) satisfies \( a^2 + b^2 = 1 \), then it lies on this unit circle. Understanding this concept is vital when analyzing circular motion, projections, and waveform studies.