Problem 32 - Precalculus : Building Concepts and Connections

Question

Determine whether the given pairs of vectors are orthogonal. $$\mathbf{v}=\langle 2,0\rangle, \mathbf{w}=\langle 0,4\rangle$$

Step-by-Step Solution

Answer

Yes, the vectors \(\mathbf{v}=\langle 2,0\rangle\) and \(\mathbf{w}=\langle 0,4\rangle\) are orthogonal since their dot product is zero.

Step 1: Understand the Concept of Orthogonality

Two vectors are orthogonal if their dot product is zero. The dot product is the sum of the multiplication of their corresponding components.

Step 2: Calculate the Dot Product

Using the formula of the dot product, multiply the corresponding elements of the two vectors and then add those products together. For the vectors \(\mathbf{v}=\langle 2,0\rangle\) and \(\mathbf{w}=\langle 0,4\rangle\), the dot product is calculated as follows: \(2*0 + 0*4 = 0.\)

Step 3: Determine if Vectors are Orthogonal

Since the dot product calculated in Step 2 is zero, hence it can be concluded that the vectors \(\mathbf{v}\) and \(\mathbf{w}\) are orthogonal.

Key Concepts

Dot Product

In vector mathematics, the dot product is a crucial operation. The dot product between two vectors is a measure of their alignment. Technically, it is computed by multiplying corresponding components of each vector and summing those products.
For vectors \(\mathbf{v}=\langle a, b\rangle\) and \(\mathbf{w}=\langle c, d\rangle\), the dot product is calculated as:

• \(a \cdot c + b \cdot d\)

A real-life analogy of the dot product is considering how much two different directions go in the same overall direction. A larger dot product means more alignment.
Notably, the dot product is a scalar quantity, meaning it results in a single number rather than a new vector.

Orthogonal Vectors

Orthogonal vectors are vectors that are perpendicular to each other. In mathematical terms, if two vectors are orthogonal, their dot product will be zero. This zero dot product means there is no overlap in their directions.
An intuitive way to think about orthogonal vectors is to imagine standing in a room with one vector pointing toward the ceiling and another pointing toward one of the walls. These vectors are perpendicular, just as orthogonal vectors are in mathematics.
Checking for orthogonality is particularly helpful in various applications, such as computer graphics and physics, where understanding angles between vectors is crucial.

Vector Mathematics

Vector mathematics involves operations like addition, subtraction, and multiplication. These operations are essential for solving problems in physics, engineering, and computer science.

• Addition: To add vectors, sum their corresponding components. \(\langle a, b \rangle + \langle c, d \rangle = \langle a+c, b+d \rangle\).
• Subtraction: For subtraction, subtract corresponding components. \(\langle a, b \rangle - \langle c, d \rangle = \langle a-c, b-d \rangle\).
• Scalar Multiplication: Multiply each component by the scalar. \(k\cdot\langle a, b \rangle = \langle ka, kb \rangle\).

Understanding these operations is vital for applications in geometry, where vectors are used to describe direction and magnitude.

Linear Algebra Concepts

Linear algebra is a branch of mathematics that studies vectors, matrices, and their operations. Within linear algebra, the orthogonality of vectors plays an important role.
This concept helps in understanding vector spaces, which are collections of vectors that can be added together and multiplied by scalars to produce another vector in the same space. In vector spaces, linear independence, basis, and dimensionality are key concepts.

• Linear Independence: A set of vectors is linearly independent if no vector can be expressed as a combination of the others.
• Basis: A basis is a set of linearly independent vectors that span the entire vector space.
• Dimensionality: This describes the number of vectors in a basis that it takes to span the space.

Grasping these concepts is fundamental for understanding systems of linear equations and transformations in mathematics.