Problem 43 - Precalculus : Building Concepts and Connections

Question

In this set of exercises, you will use vectors and dot products to study real- world problems. In a new video game, Mario and Luigi are at positions defined by the vectors \langle 10,3\rangle and \(\langle x, 15\rangle .\) What must be the value of \(x\) so that their position vectors are orthogonal?

Step-by-Step Solution

Answer

The value of \(x\) that makes the position vectors of Mario and Luigi orthogonal is \(x=-4.5\).

Step 1: Write Down the Given Vectors

We know that Mario's position is given by the vector \(\langle 10, 3\rangle\) and Luigi’s position is defined by the vector \(\langle x, 15\rangle\).

Step 2: Dot Product of Vectors

The dot product of two vectors \(\langle a, b\rangle\) and \(\langle c, d\rangle\) is given by \(ac + bd\). Applying this here, we get that the dot product of the given vectors is \(10*x + 3*15\).

Step 3: Solve for X

For the two vectors to be orthogonal, their dot product must be zero. So, set the dot product equal to zero and solve for \(x\). This yields the equation \(10*x + 45 = 0\). Solving for \(x\) gives us \(x=-4.5\).

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a powerful mathematical tool used to determine the relationship between two vectors in geometry and physics. Essentially, the dot product calculates the product of the magnitudes of two vectors and the cosine of the angle between them. In precalculus and higher-level mathematics, the dot product is defined algebraically for two-dimensional vectors \( (a, b) \) and \( (c, d) \) as \( ac + bd \).

For the vectors to be orthogonal, which means they meet at a right angle, their dot product must be zero because the cosine of 90 degrees is zero. In the exercise, by finding the value of \( x \) that ensures the dot product is zero, you are essentially determining the condition required for Mario and Luigi's position vectors to be perpendicular to each other. This concept not only applies to video game positions but to many real-world scenarios where orthogonality plays a role, such as calculating forces, optimizing designs, or analyzing movement.

Vector

In precalculus, a vector is a mathematical object that possesses both a magnitude and a direction. Unlike a scalar, which only has magnitude, vectors are essential for representing quantities such as force, velocity, and displacement in physics. A vector in two dimensions is usually written in the form \( \langle a, b \rangle \) where \( a \) and \( b \) are the components that indicate the vector's magnitude in the horizontal and vertical directions, respectively.

They can be represented graphically as arrows pointing from one point to another or algebraically as ordered pairs, like in our exercise. Each component of the vector contributes to its overall direction and magnitude. By manipulating vectors algebraically, as done by finding orthogonal vectors through dot product, we simplify complex geometric problems into an easily solvable format for practical applications.

Precalculus

Precalculus is an intermediate level of mathematical education that prepares students for calculus. It encompasses a review of algebra, geometry, and functions, and introduces new concepts such as vectors, limits, and trigonometric functions. Precalculus helps develop the foundational skills necessary to understand and handle calculus problems, which is essential for disciplines involving advanced mathematics.

Through exercises such as finding orthogonal vectors using dot product, precalculus students practice reasoning and analytical skills. They engage with abstract concepts in a tangible way, which is vital to success in calculus and related fields. The problem in our exercise encourages problem-solving and critical thinking, as students are asked to identify conditions under which vectors behave in a specific manner — skills that are crucial across scientific and engineering domains.