Problem 20 - Algebra and Trigonometry

Question

Find \(b\) so that the vectors \(\mathbf{v}=\mathbf{i}+\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+b \mathbf{j}\) are orthogonal.

Step-by-Step Solution

Answer

The value of \(b\) is \(-1\).

Step 1: Understand orthogonal vectors

Two vectors are orthogonal if their dot product is zero. The dot product of \(\textbf{v}\) and \(\textbf{w}\) must be calculated and set equal to zero.

Step 2: Write the dot product formula

The dot product of \(\textbf{v}\) and \(\textbf{w}\) is computed as follows: \[ \textbf{v} \bullet \textbf{w} = (1)\times(1) + (1)\times(b) \]

Step 3: Set up the equation

Set the dot product equal to zero to find the value of \(b\): \[ 1 + b = 0 \]

Step 4: Solve for \(b\)

Solve the equation \(1 + b = 0\) to find \(b\): \[ b = -1 \]

Key Concepts

dot product

The dot product is a fundamental operation in vector algebra. It takes two vectors and returns a scalar value. This value is calculated by multiplying corresponding components of the vectors and then summing those products. For two vectors \(\textbf{a} = [a_1, a_2, ..., a_n]\) and \(\textbf{b} = [b_1, b_2, ..., b_n]\), the dot product is given by: \[ \textbf{a} \bullet \textbf{b} = a_1 \times b_1 + a_2 \times b_2 + ... + a_n \times b_n \] The dot product has many applications such as finding the angle between two vectors, determining if vectors are orthogonal, and calculating projections. Orthogonal vectors have a dot product of zero, indicating that they are perpendicular to each other.

vector algebra

Vector Algebra is a branch of mathematics that deals with vectors and the rules for manipulating them. Vectors have both magnitude and direction, and they can be added, subtracted, and multiplied using operations such as the dot product and cross product.
In vector algebra, vectors can be represented in different forms such as Cartesian coordinates. For instance, vector \(\textbf{v} = \textbf{i} + \textbf{j}\) represents a vector in 2D space with components 1 along the x-axis and 1 along the y-axis.
Basic operations in vector algebra include:

• Vector Addition: Combine corresponding components.
• Vector Subtraction: Subtract corresponding components.
• Scalar Multiplication: Multiply each component by a scalar.
• Dot Product: Multiply corresponding components and sum them.

Understanding these operations is crucial for solving problems in physics, engineering, and computer graphics.

solving equations

Solving equations is a key part of finding solutions in linear algebra and vector problems. In our example, we needed to find \(b\) such that two vectors are orthogonal. This involved setting up and solving a simple linear equation.
Here are the steps we followed:

• Compute the dot product: \(\textbf{v} \bullet \textbf{w} = 1 + b\)
• Set the dot product equation equal to zero: \(1 + b = 0\)
• Solve for \(b\) by isolating the variable: \(b = -1\)

These steps illustrate a basic process in solving linear equations: manipulate the equation to isolate the unknown variable. This skill is useful across many areas of mathematics and science.

linear algebra

Linear Algebra is the study of vectors, vector spaces, and linear transformations. This field is fundamental to many areas such as physics, computer science, and engineering. It provides tools for modeling and solving a variety of problems.
Core concepts in linear algebra include:

• Vectors and Vector Spaces: Collections of vectors that adhere to specific rules.
• Matrices: Rectangular arrays of numbers representing linear transformations.
• Determinants: Scalars derived from matrices, used in solving systems of linear equations.
• Eigenvalues and Eigenvectors: Special numbers and vectors associated with matrices.

In our example, understanding orthogonal vectors and the dot product is a fundamental application of linear algebra. Recognizing when vectors are perpendicular by using the dot product helps solve geometric and physical problems.