Question

Find a vector perpendicular to both \(\mathbf{i}+\mathbf{j}\) and \(\mathbf{i}-2 \mathbf{k}\).

Short answer

The vector is \(-2\textbf{j} - \textbf{k}\).

Step-by-step solution

Define the Vectors

Identify the given vectors. Let \(\textbf{A} = \textbf{i} + \textbf{j}\) and \(\textbf{B} = \textbf{i} - 2\textbf{k}\).

Use the Cross Product

To find a vector perpendicular to both \(\textbf{A} \) and \(\textbf{B}\), we use the cross product \( \textbf{A} \times \textbf{B} \).

Set up the Determinant

Set up the cross product determinant using the unit vectors \(\textbf{i}, \textbf{j}, \textbf{k}\): \(\begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \text{ 1 } & \text{1} & \text{0} \ \text{1} & \text{0} & -2 \ \end{vmatrix} \).

Calculate the Determinant

Compute the determinant. \(\textbf{A} \times \textbf{B} = \textbf{i} \begin{vmatrix} 1 & 0 \ 0 & -2 \end{vmatrix} - \textbf{j} \begin{vmatrix} 1 & -2 \ 1 & 0 \end{vmatrix} + \textbf{k} \begin{vmatrix} 1 & 1 \ 1 & 0 \end{vmatrix} \).

Solve the Determinants

Solve for each cofactor: \(\textbf{i}(0 \times -2 - 1 \times 0) - \textbf{j}(1 \times 0 - 1 \times -2) + \textbf{k}(1 \times 0 - 1 \times 1) \).

Simplify the Result

Simplify to get the resulting vector: \(\textbf{0i} - \textbf{2j} - \textbf{1k} \), or \(-2\textbf{j} - \textbf{k}\).

Key concepts

Cross Product

When you need to find a vector that is perpendicular to two given vectors, the cross product is your go-to operation. It's represented by \(\textbf{A} \times \textbf{B}\), where \(\textbf{A}\) and \(\textbf{B}\) are the two vectors. The cross product yields another vector that is perpendicular to the original vectors. This is particularly useful in problems involving physics, engineering, and computer graphics.

The resulting vector's direction can be determined using the right-hand rule. Point your index finger of the right hand in the direction of \(\textbf{A}\), and your middle finger in the direction of \(\textbf{B}\). Your thumb will point in the direction of \(\textbf{A} \times \textbf{B}\). Simply put:

• \textbf{A} \times \textbf{B}\ is perpendicular to both \(\textbf{A}\) and \(\textbf{B}\).
• Use the right-hand rule for the direction.

Sometimes students struggle to understand the concept without visualization, so always remember the right-hand rule and its application in 3D space.

Determinant

To calculate the cross product, we use the determinant of a matrix formed by the unit vectors \(\textbf{i}, \textbf{j}, \textbf{k}\) and the components of the vectors \(\textbf{A}\) and \(\textbf{B}\). Setting up this matrix helps in systematically carrying out the operation.

In our case, the determinant looks like this:
\(\begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ 1 & 1 & 0 \ 1 & 0 & -2 \ \end{vmatrix}\).

Each component of the resulting vector will be calculated using the minor determinants. For example, the \(\textbf{i}\) component is found by ignoring the row and column of \(\textbf{i}\), then calculating the determinant of the remaining 2x2 matrix. Each cofactor is similarly found, giving us the perpendicular vector.

• Set up a 3x3 determinant with vectors and unit vectors.
• Calculate minor determinants for each component using 2x2 matrices.

Understanding determinants is essential for proper computation in more complex vector operations.

Vector Operations

Vector operations like addition, subtraction, and cross product help us manipulate vectors to get the desired results. In this exercise, we used the cross product, but understanding other operations is equally important:

• Addition: Combine corresponding components.
• Subtraction: Subtract corresponding components.
• Cross Product: Finds a vector perpendicular to the original ones.

Basics of vector operations come handy in various fields requiring spatial understanding, such as physics and computer graphics. Vector operations allow us to describe and manipulate physical quantities like forces, velocities, and positions in space. Breaking down complex operations into smaller, manageable steps can make these concepts easier to understand.

Unit Vectors

Unit vectors, usually denoted as \(\textbf{i}, \textbf{j}, \textbf{k}\), are vectors with a magnitude of one. These unit vectors represent the standard axes in 3D space:

• \textbf{i}\: Points along the x-axis
• \textbf{j}\: Points along the y-axis
• \textbf{k}\: Points along the z-axis

They form the basis for constructing and deconstructing other vectors. Utilizing unit vectors makes it easier to set up matrices and perform vector operations like the cross product. Understanding unit vectors helps you grasp more complex vector quantities and transform operations to simpler, element-wise calculations. This exercise shows you how we use unit vectors to set up a cross product determinant, making difficult 3D vector manipulations more approachable.

Always remember that understanding and visualizing unit vectors can simplify your work significantly.