Problem 17 - APEX Calculus

Question

In Exercises 17-20, a vector \(\vec{v}\) is given. Give two vectors that are orthogonal to \(\vec{v}\). \(\vec{v}=\langle 4,7\rangle\)

Step-by-Step Solution

Answer

Two orthogonal vectors are \(\langle 7, -4 \rangle\) and \(\langle 1, -\frac{4}{7} \rangle\).

Step 1: Define Orthogonal Vectors

Two vectors are orthogonal if their dot product equals zero. Therefore, given a vector \(\vec{v} = \langle 4, 7 \rangle\), we need to find vectors \(\vec{u} = \langle a, b \rangle\) such that the dot product \(4a + 7b = 0\).

Step 2: Determine the First Orthogonal Vector

To find the first orthogonal vector, we can set \(a = 7\) and then solve for \(b\) in the equation \(4a + 7b = 0\). Substituting gives \(4(7) + 7b = 0\), which simplifies to \(28 + 7b = 0\). Solving for \(b\), we get \(b = -4\). Thus, the first orthogonal vector is \(\langle 7, -4 \rangle\).

Step 3: Determine the Second Orthogonal Vector

To find a second orthogonal vector, we can set \(a = 1\) and solve for \(b\). Substituting into the equation \(4a + 7b = 0\), we get \(4(1) + 7b = 0\), which simplifies to \(4 + 7b = 0\). Solving for \(b\), we find \(b = -\frac{4}{7}\). Thus, the second orthogonal vector is \(\langle 1, -\frac{4}{7} \rangle\).

Key Concepts

Dot Product

The dot product is a fundamental operation in vector arithmetic used to determine the relationship between two vectors. It involves multiplying the corresponding components of two vectors and summing the results. For two-dimensional vectors \( \vec{a} = \langle a_1, a_2 \rangle \) and \( \vec{b} = \langle b_1, b_2 \rangle \), the dot product is calculated as:\[\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2\]This calculation tells us the scalar product of the two vectors, which is important for various tasks, such as finding if two vectors are orthogonal. Vectors are orthogonal when their dot product is zero. This means they are perpendicular to each other in space, making the dot product crucial in determining this special relationship.

Vector Arithmetic

Vector arithmetic is the art of manipulating vectors through various operations such as addition, subtraction, and scalar multiplication. It forms the backbone of many vector operations in mathematics and physics.

• Addition: To add vectors \( \vec{a} = \langle a_1, a_2 \rangle \) and \( \vec{b} = \langle b_1, b_2 \rangle \), simply add their corresponding components: \( \langle a_1 + b_1, a_2 + b_2 \rangle \).
• Subtraction: Similarly, vector subtraction involves subtracting the components: \( \langle a_1 - b_1, a_2 - b_2 \rangle \).
• Scalar Multiplication: To multiply a vector by a scalar \(c\), multiply each component by \(c\): \( \langle c \, a_1, c \, a_2 \rangle \).

Understanding vector arithmetic allows for the manipulation and analysis of vector quantities, providing insights into geometric and physical relationships.

Linear Algebra

Linear Algebra is a branch of mathematics that studies vectors, vector spaces, and linear transformations. It provides the tools and frameworks to analyze systems of linear equations and vector spaces systematically. A key component of linear algebra is the concept of a vector space. This is a collection of vectors that can be scaled and added together while still remaining within the system. Vectors in a vector space can be explored through operations like the dot product, discovered earlier, or through linear combinations to form new vectors. Linear algebra finds its use in various fields such as computer graphics, where vector spaces represent coordinate systems. In physics, it helps model forces and movements. Understanding concepts in linear algebra, like orthogonality and dot product as shown in the exercise, lays a fundamental groundwork for more complex operations and technologies.