Problem 4 - Algebra and Trigonometry

Question

For the given vectors a and b, find the cross product \(\mathbf{a} \times \mathbf{b}\). $$ \mathbf{a}=\langle 0,-4,1\rangle, \quad \mathbf{b}=\langle 1,1,-2\rangle $$

Step-by-Step Solution

Answer

The cross product is \( \langle 7, 2, 4 \rangle \).

Step 1: Write the Cross Product Formula

The cross product of two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \) is given by \( \mathbf{a} \times \mathbf{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle \).

Step 2: Write the equation in standard form

Rearrange and simplify the equation.

Step 3: Apply the solution method

Use factoring, quadratic formula, substitution, or other methods.

Step 4: Verify the solution(s)

Check solutions in the original equation.

Step 5: State the final answer

List all valid solutions.

Step 6: Conclude with the answer

The cross product is \( \langle 7, 2, 4 \rangle \).

Key Concepts

Vector Algebra

Vector algebra is an essential part of mathematics that deals with vector quantities. Unlike scalar quantities, vectors have both a magnitude and a direction. In vector algebra, we perform operations with these vectors to understand their relationships and properties.
Some of the common operations you might encounter include addition, subtraction, scalar multiplication, dot product, and cross product. These operations allow us to solve problems related to force, velocity, displacement, and many other physical quantities.
The cross product, specifically, is a vector operation that results in another vector that is perpendicular to the plane formed by the initial two vectors. This makes it particularly useful in computing areas of parallelograms and finding unit vectors normal to a plane.

Three-Dimensional Vectors

Three-dimensional vectors are a basic representation of quantities in three-dimensional space. They are expressed in terms of the three coordinate axes: x, y, and z. A vector \(\mathbf{a}\) is written as \langle a_1, a_2, a_3 \rangle\ where the components \(a_1\), \(a_2\) and \(a_3\) represent its projection along the x, y, and z axes, respectively.
Each component describes how far the vector extends in the respective direction. Understanding three-dimensional vectors is crucial when dealing with problems in physics and engineering, such as calculating forces or trajectories.
Vectors in three-dimensional space often require operations that involve all three components, like the cross product, which emphasizes how these vectors can interact within space to produce meaningful results.

Vector Operations

Vector operations are methods of manipulating two or more vectors to produce new vectors as results. These operations include both basic arithmetic-like addition and subtraction, as well as more complex operations like the dot product and the cross product.

• Addition and Subtraction: Combine vectors by adding or subtracting corresponding components.
• Scalar Multiplication: Multiply each component of the vector by a scalar (a real number).
• Dot Product: Produces a scalar and is useful for finding the angle between vectors.
• Cross Product: Produces another vector, perpendicular to the original vectors, used extensively in three-dimensional calculations.

The cross product formula, as described in the exercise, computes the resulting vector in terms of its components:\[\mathbf{a} \times \mathbf{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle\]
Practicing these operations helps in visualizing spatial problems and understanding geometric and physical phenomena.