Question

Letting \(i, j, k\) be first-order units in three-dimensional space, determine the combinatory product of \(2 i+3 j-4 k\), \(3 i-i+k \cdot i+2 i-k\)

Short answer

Answer: The cross product is \(-3i + 18j - 15k\).

Step-by-step solution

Arrange the vectors in matrix form

First, let's simplify the second vector: \(3i - i + k \cdot i + 2i - k = (3 - 1 + 1 + 2)i - k = 5i - k\). Now, arrange the two given vectors in matrix form with unit vectors \(i, j, k\) in the first row and the components of the vectors in the next two rows: | i | j | k | | --- | --- | --- | | 2 | 3 | -4 | | 5 | 0 | -1 |

Calculate the determinants of the submatrices

In order to find the cross product, we need to find the determinants of the 2x2 submatrices for each of the unit vectors \(i, j, k\). For \(i\): ``` | 3 | -4 | | --- | --- | | 0 | -1 | ``` Determinant = \((3 \times (-1)) - (0 \times (-4)) = -3\) For \(j\): ``` | 2 | -4 | | --- | --- | | 5 | -1 | ``` Determinant = \((2 \times (-1)) - (5 \times (-4)) = -2 + 20 = 18\) For \(k\): ``` | 2 | 3 | | --- | --- | | 5 | 0 | ``` Determinant = \((2 \times 0) - (5 \times 3) = 0 - 15 = -15\)

Find the cross product using the determinants

Now we can find the cross product by combining the determinants with their corresponding unit vectors: Cross Product = \((-3)i + (18)j + (-15)k\) So the combinatory product of the given vectors is \(-3i + 18j - 15k\).

Key concepts

Cross Product

The cross product is an important operation in vector mathematics that occurs between two vectors in three-dimensional space. It is unique because unlike the dot product, the cross product yields a vector rather than a scalar. This vector is orthogonal, or perpendicular, to both of the original vectors involved. This property makes it highly useful in physics and engineering for finding the direction of a force or a magnetic field.

To compute the cross product of two vectors, say \(\mathbf{A} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k}\) and \(\mathbf{B} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k}\), we use the formula:
\[\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix}\]
This determinant expands into smaller sub-determinants that involve the components of the original vectors, which we calculate for each of \(\mathbf{i}, \mathbf{j}, \mathbf{k}\). In this determinant, each component's sub-determinant formula corresponds to a specific unit vector.

• \(\mathbf{i}: (a_2 b_3 - a_3 b_2)\)
• \(\mathbf{j}: - (a_1 b_3 - a_3 b_1)\)
• \(\mathbf{k}: (a_1 b_2 - a_2 b_1)\)

The result is a new vector, with each component determined by a 2x2 sub-determinant.

Three-Dimensional Space

Three-dimensional space is the environment we usually visualize when thinking about the physical world. It involves three directions: length, width, and height. Each point in this space is defined by three coordinates, commonly expressed as \((x, y, z)\) or through unit vectors \(\mathbf{i}, \mathbf{j}, \mathbf{k}\), commonly associated with the x, y, and z axes respectively.

In vector mathematics, operations such as addition, subtraction, and products (like the dot and cross product) require understanding and manipulating these vectors in 3D space. Vectors in three-dimensional space can have both direction and magnitude, leading to various applications in physics, mathematics, and even computer graphics.

Visualizing vectors in 3D space can be challenging, but imagining them as arrows that point in the direction of the vector with length proportional to the magnitude can help. Operations like the cross product become a tool to understand and manipulate spaces when systems involve objects or paths not limited to just forwards and backwards, but also upwards, downwards and sideways.

Determinants

Determinants are a fundamental tool in linear algebra, particularly when dealing with matrices. In the context of the cross product in vector mathematics, determinants help simplify complex calculations required for finding the vector's perpendicular direction in three-dimensional space.

Technically, a determinant is a scalar value derived from a square matrix. For a 2x2 matrix, the determinant is calculated as follows:\[ \begin{vmatrix} a & b \ c & d \end{vmatrix} = a \times d - b \times c\]
For a 3x3 matrix used in the cross product, the calculation involves breaking it down into multiple 2x2 determinants, as explored in the previous sections. Each of these smaller determinants corresponds to one of the unit vectors \(\mathbf{i}, \mathbf{j}, \mathbf{k}\).

Determinants help us understand concepts such as:

• The area of parallelograms (2D) or volume of parallelepipeds (3D) made by vectors.
• The invertibility of a matrix—non-zero determinant implies invertibility.

Through their use in deriving the cross product, determinants play a crucial role in simplifying vector operations in multidimensional spaces.