Problem 28 - Precalculus : Building Concepts and Connections

Question

Determine whether the given pairs of vectors are orthogonal. $$\mathbf{v}=\langle-5,2\rangle, \mathbf{w}=\langle 4,-10\rangle$$

Step-by-Step Solution

Answer

The vectors \(\mathbf{v}=\langle-5,2\rangle\) and \(\mathbf{w}=\langle 4,-10\rangle\) are not orthogonal.

Step 1: Identify the given vectors

The given vectors are recognisable as \(\mathbf{v}=\langle-5,2\rangle\) and \(\mathbf{w}=\langle 4,-10\rangle\).

Step 2: Calculate the dot product

The dot product of two vectors is calculated by multiplying their corresponding components and then adding those products. The dot product of \(\mathbf{v}\) and \(\mathbf{w}\) is given as \(\mathbf{v} \cdot \mathbf{w} = (-5*4)+(2*-10)\).

Step 3: Check if the dot product is zero

After calculating, we find that \(\mathbf{v} \cdot \mathbf{w} = -20 -20 = -40\).

Step 4: Determine if vectors are orthogonal

If the dot product is zero, the vectors are orthogonal. Since the dot product in this instance is not zero, we can conclusively state that these vectors are not orthogonal.

Key Concepts

Dot Product

Understanding the dot product is fundamental when working with vectors. In essence, the dot product is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. This operation is also known as the scalar product because the result is a scalar.

Let's break down how to compute the dot product. Suppose we have two vectors, \(\mathbf{a}\) with components \(\mathbf{a} = \langle a_1, a_2, \ldots, a_n \rangle\) and \(\mathbf{b}\) with components \(\mathbf{b} = \langle b_1, b_2, \ldots, b_n \rangle\). The dot product of \(\mathbf{a}\) and \(\mathbf{b}\) is found by multiplying their corresponding components and summing those products:

\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n\]

This calculation leads to a variety of applications. One notable use of the dot product is to determine the angle between two vectors. If the dot product is zero, it indicates that the vectors are perpendicular, or ‘orthogonal’ to each other.

Vector Components

Vector components are the building blocks of vectors in multidimensional spaces. A vector in two-dimensional space is represented by a pair of numbers, which are known as its components. These numbers convey the direction and magnitude along each axis of the coordinate system.

Take a vector \(\mathbf{v}\), which can be written in component form as \(\mathbf{v} = \langle v_x, v_y \rangle\). Here, \(v_x\) is the component along the x-axis and \(v_y\) is the component along the y-axis. When we extend this idea to three dimensions, the vector includes a third component, \(v_z\), which represents its magnitude along the z-axis.

Breaking down vectors into their components is crucial for vector operations, including addition, subtraction, and scaling, as well as for finding the dot product and cross product. By understanding vector components, one can easily visualize and solve vector-related problems in geometry and physics.

Orthogonality Condition

The concept of orthogonality is very important in vector algebra, especially in geometrical contexts. Two vectors are said to be orthogonal if they meet at a right angle (90 degrees) to each other. One of the simplest ways to verify if two vectors are orthogonal is by using their dot product.

The orthogonality condition states that for two vectors to be orthogonal, their dot product must equal zero:
\[\mathbf{a} \cdot \mathbf{b} = 0\]

When you have the vector components, you can apply the dot product calculation we discussed earlier. If the result is zero, you can be confident that the vectors are indeed orthogonal. This condition is widely used across various fields of mathematics and science, including computer graphics, engineering, and physics, wherever perpendicular vectors play a critical role.