Problem 1 - Algebra and Geometry

Question

$$ \text { Solve the equations } \mathbf{x}+(\mathbf{x} \cdot \mathbf{i}) \mathbf{i}=\mathbf{j} . \text { and } \mathbf{x}+(\mathbf{x} \times \mathbf{i})=\mathbf{j} $$

Step-by-Step Solution

Answer

1. \( \mathbf{x} = \mathbf{j} \); 2. \( \mathbf{x} = -\frac{1}{2} \mathbf{j} + \frac{1}{2} \mathbf{k} \).

Step 1: Understand the Given Equation

The given vector equations involve operations with the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \). The first equation is \( \mathbf{x} + (\mathbf{x} \cdot \mathbf{i}) \mathbf{i} = \mathbf{j} \), where \( \mathbf{x} \cdot \mathbf{i} \) is the dot product resulting in a scalar. The second equation is \( \mathbf{x} + (\mathbf{x} \times \mathbf{i}) = \mathbf{j} \), where \( \mathbf{x} \times \mathbf{i} \) is the cross product resulting in a vector.

Step 2: Define the Vector \( \mathbf{x} \)

Assume the vector \( \mathbf{x} = x_1 \mathbf{i} + x_2 \mathbf{j} + x_3 \mathbf{k} \). This allows us to break down the problem into its component parts using unit vectors \( \mathbf{i}, \mathbf{j}, \text{ and } \mathbf{k} \).

Step 3: Solve the First Equation

The first equation is \( \mathbf{x} + (\mathbf{x} \cdot \mathbf{i}) \mathbf{i} = \mathbf{j} \). Substituting \( \mathbf{x} = x_1 \mathbf{i} + x_2 \mathbf{j} + x_3 \mathbf{k} \), the dot product \( \mathbf{x} \cdot \mathbf{i} = x_1 \). The equation becomes \( (x_1 + x_1) \mathbf{i} + x_2 \mathbf{j} + x_3 \mathbf{k} = \mathbf{j} \). Equating components, we find \( 2x_1 = 0 \), \( x_2 = 1 \), and \( x_3 = 0 \), giving the solution \( \mathbf{x} = 0 \mathbf{i} + 1 \mathbf{j} + 0 \mathbf{k} \).

Step 4: Solve the Second Equation

For the second equation \( \mathbf{x} + (\mathbf{x} \times \mathbf{i}) = \mathbf{j} \), the cross product \( \mathbf{x} \times \mathbf{i} = (x_2 \mathbf{k} - x_3 \mathbf{j}) \). Substituting into the equation, we have \( x_1 \mathbf{i} + x_2 \mathbf{j} + x_3 \mathbf{k} + x_2 \mathbf{k} - x_3 \mathbf{j} = \mathbf{j} \). Simplifying by component: \( x_1 \mathbf{i} + (x_2 - x_3) \mathbf{j} + (x_3 + x_2) \mathbf{k} = \mathbf{j} \). Equating, \( x_1 = 0 \), \( x_2 - x_3 = 1 \), \( x_3 + x_2 = 0 \). Solving these, we get \( x_2 = -\frac{1}{2} \) and \( x_3 = \frac{1}{2} \). So, \( \mathbf{x} = 0 \mathbf{i} - \frac{1}{2} \mathbf{j} + \frac{1}{2} \mathbf{k} \).

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It combines two vectors to produce a scalar quantity. Unlike other vector operations, the result of a dot product is not a vector.
To compute the dot product of two vectors, you multiply their corresponding components and sum the results:

• If you have two vectors, \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \), the dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as \( a_1b_1 + a_2b_2 + a_3b_3 \).
• This operation measures how much one vector extends in the direction of another. It's very useful in physics to find the work done when a force is applied to an object moving in a straight line.

Cross Product

In contrast to the dot product, the cross product of two vectors results in another vector. This vector is perpendicular to both of the original vectors, making it useful for determining the orientation in space.
To compute the cross product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \):

• For \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \), the cross product \( \mathbf{a} \times \mathbf{b} \) is a vector given by:
• \[ (a_2b_3 - a_3b_2)\mathbf{i} + (a_3b_1 - a_1b_3)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k} \]
• This vector's magnitude corresponds to the area of the parallelogram spanned by \( \mathbf{a} \) and \( \mathbf{b} \).

It's critical for applications in three-dimensional geometry and determining torque.

Vector Components

Understanding vector components is essential in breaking down complex vector operations. Vectors can be expressed in terms of unit vectors, which are typically labeled as \( \mathbf{i}, \mathbf{j}, \) and \( \mathbf{k} \) in three dimensions. These unit vectors represent directions along the x, y, and z axes, respectively.
To express a vector \( \mathbf{x} \) in terms of its components, you write:

• \( \mathbf{x} = x_1 \mathbf{i} + x_2 \mathbf{j} + x_3 \mathbf{k} \)
• The scalars \( x_1, x_2, \) and \( x_3 \) are the vector components along the x, y, and z axes respectively.

These components allow for simplifying operations like the dot and cross products by aligning the vectors with unit vector coordinates, making calculations straightforward by handling them step by step.