Problem 83 - A Graphical Approach to Precalculus with Limits

Question

Determine whether each pair of vectors is orthogonal. $$\sqrt{5} i-2 j,-5 i+2 \sqrt{5} j$$

Step-by-Step Solution

Answer

The vectors are not orthogonal since their dot product is \(-9\sqrt{5}\), which is not zero.

Step 1: Define Orthogonality

Two vectors are orthogonal if their dot product is zero. For vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), the dot product is calculated as \( a_1 b_1 + a_2 b_2 \).

Step 2: Identify Components

The given vectors are \( \mathbf{a} = \sqrt{5}\mathbf{i} - 2\mathbf{j} \) and \( \mathbf{b} = -5\mathbf{i} + 2\sqrt{5}\mathbf{j} \). So, the components are: \( a_1 = \sqrt{5}, a_2 = -2, b_1 = -5, b_2 = 2\sqrt{5}.\)

Step 3: Compute the Dot Product

Calculate the dot product of \( \mathbf{a} \) and \( \mathbf{b} \) by substituting the components into the dot product formula: \[a_1 b_1 + a_2 b_2 = \sqrt{5}(-5) + (-2)(2\sqrt{5})\]Simplifying gives:\[-5\sqrt{5} - 4\sqrt{5} = -9\sqrt{5}.\]

Step 4: Determine Orthogonality of Vectors

The dot product is \(-9\sqrt{5}\), which is not zero. Thus, the vectors \( \sqrt{5} \mathbf{i} - 2 \mathbf{j} \) and \( -5 \mathbf{i} + 2 \sqrt{5} \mathbf{j} \) are not orthogonal.

Key Concepts

Dot Product

In precalculus, the dot product is a fundamental operation used to find the relationship between two vectors. The dot product, also known as the scalar product, is computed from two vectors, say \( \mathbf{a} \) and \( \mathbf{b} \). The formula to find the dot product in two dimensions is given by:

• \( a_1 b_1 + a_2 b_2 \)

The notation \( a_1, a_2, b_1, b_2 \) represents the respective components of the vectors \( \mathbf{a} \) and \( \mathbf{b} \). It is important to remember that the result of a dot product is always a scalar, not another vector.

One key concept related to the dot product is orthogonality. Two vectors are orthogonal, which means they are perpendicular, if their dot product equals zero. This is a quick way to check if vectors meet at a right angle.

In the given problem, by calculating \( -9\sqrt{5} \), we see that the vectors have a non-zero dot product, confirming they are not orthogonal.

Vectors

Vectors are mathematical objects that have both magnitude and direction. In the context of precalculus, vectors are often represented in terms of the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) which correspond to the x and y components in a two-dimensional space. A vector \( \mathbf{v} \) can be written as:

• \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} \)

Here, \( v_1 \) and \( v_2 \) are the vector's horizontal and vertical components respectively.

Understanding vectors is crucial because they provide a way to encapsulate information about an object’s direction and speed. Vectors are not just limited to two dimensions; they can represent physics problems in multiple dimensions, but in precalculus, we tend to deal with two dimensions.

Vectors allow us to perform operations such as addition, subtraction, and computation of dot products, which help determine geometric and algebraic relationships between them. In this exercise, identifying the vector components correctly was essential for calculating the dot product.

Precalculus Concepts

Precalculus is the branch of mathematics that prepares students for calculus. It encompasses a variety of topics including functions, complex numbers, trigonometry, and vectors. Within precalculus, vectors are often introduced in a geometric context. Understanding vectors and operations like the dot product are crucial for solving complex calculus problems in later studies.

Precalculus helps students bridge the gap between algebra and calculus by introducing concepts that rely on logical and spatial thinking. This includes using vectors to break down physical quantities into manageable mathematical terms. By doing so, students get a feel for the mathematical landscape they will navigate in calculus.

This exercise is an example of how precalculus concepts help in analyzing vector properties like orthogonality. By learning these fundamental concepts, you set the stage for more advanced studies in mathematics and related fields.