Building Perpendicularity: Exploring the Cross Product’s Role in Construction and Design

One application of cross product of vector
byu/DigitalSplendid inLinearAlgebra

Disclaimer: This article was created with the assistance of an AI language model and is intended for informational purposes only. Please verify any technical details before implementation.

Yes, your example provides a conceptual application of the cross product. The cross product of two vectors in 3D space produces a vector that is perpendicular to the plane formed by those two vectors. This property is particularly useful in understanding and solving problems involving perpendicularity and orientation. Let’s analyze your example:

Explanation:

1. Vectors for Land Dimensions:
   Represent the length and width of the land as vectors:

• ( \mathbf{L} = \langle 10, 0, 0 \rangle ) (along the (x)-axis, representing length)
• ( \mathbf{W} = \langle 0, 10, 0 \rangle ) (along the (y)-axis, representing width)

1. Cross Product Application:
   The cross product ( \mathbf{L} \times \mathbf{W} ) gives a vector perpendicular to both ( \mathbf{L} ) and ( \mathbf{W} ).

• Using the formula:
  [
  \mathbf{L} \times \mathbf{W} = \begin{vmatrix}
  \mathbf{i} & \mathbf{j} & \mathbf{k} \
  10 & 0 & 0 \
  0 & 10 & 0
  \end{vmatrix} = \langle 0, 0, 100 \rangle
  ]
• This result, ( \langle 0, 0, 100 \rangle ), points along the (z)-axis, which is perpendicular to the plane defined by the land’s length and width.

1. Angle Computation:

• The cross product inherently ensures that the resulting vector is perpendicular to both input vectors, meaning the angle between the resulting vector and each of the original vectors is (90^\circ).
• This property can be used to define the orientation of a building or any structure relative to the ground (as in your example).

Real-World Application:

The cross product is indeed used in practical scenarios like:

• Engineering and Construction: Determining perpendicular directions for setting up walls, beams, or structural elements relative to a plane.
• Physics: Computing torque, where the cross product determines the direction of the rotational force.
• Computer Graphics: Calculating surface normals for lighting and shading.

Conclusion:

Your example of determining a (90^\circ) angle for constructing a building is a valid and practical application of the cross product’s fundamental property of perpendicularity. It demonstrates how mathematical concepts translate into real-world utility in design and construction.