Okay, here's how to find a perpendicular vector in 3D.
The most common method to find a vector perpendicular to two given vectors in 3D space is by using the cross product.
To find a vector perpendicular to two other 3D vectors, you use the cross product operation. This is a fundamental concept in linear algebra and vector calculus, particularly useful in physics and computer graphics.
Based on the provided information, Given two vectors in 3D, the cross product v × w is a vector that is perpendicular to both of them.
Let's break down how this works.
Using the Cross Product
If you have two vectors, say v = (v₁, v₂, v₃) and w = (w₁, w₂, w₃), their cross product, v × w, results in a new vector that is orthogonal (perpendicular) to both v and w.
The cross product can be calculated using a determinant of a matrix involving the standard basis vectors i, j, and k:
v × w = $\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ v_1 & v_2 & v_3 \ w_1 & w_2 & w_3 \end{vmatrix}$
Expanding this determinant gives the components of the resulting perpendicular vector:
v × w = $(v_2 w_3 - v_3 w_2)\mathbf{i} - (v_1 w_3 - v_3 w_1)\mathbf{j} + (v_1 w_2 - v_2 w_1)\mathbf{k}$
This means the resulting vector is:
(v₂w₃ - v₃w₂, v₃w₁ - v₁w₃, v₁w₂ - v₂w₁)
Cross Product Formula Explained
Here's a simple breakdown of the components:
- X-component: (v₂ w₃) - (v₃ w₂)
- Y-component: (v₃ w₁) - (v₁ w₃) (Note the sign change or order reversal compared to a standard determinant expansion)
- Z-component: (v₁ w₂) - (v₂ w₁)
Alternatively, you can remember the positive terms for each component:
| Component | Formula |
|---|---|
| i (X) | v₂w₃ - v₃w₂ |
| j (Y) | -(v₁w₃ - v₃w₁) |
| k (Z) | v₁w₂ - v₂w₁ |
Example: Finding a Vector Perpendicular to Two Vectors
Let's find a vector perpendicular to v = (1, 2, 3) and w = (4, 5, 6).
Using the formula:
- X-component: (2 6) - (3 5) = 12 - 15 = -3
- Y-component: (3 4) - (1 6) = 12 - 6 = 6
- Z-component: (1 5) - (2 4) = 5 - 8 = -3
So, the vector perpendicular to both v and w is (-3, 6, -3).
You can verify this by taking the dot product of the resulting vector with v and w. If the dot product is zero, the vectors are perpendicular.
- (-3, 6, -3) ⋅ (1, 2, 3) = (-3 1) + (6 2) + (-3 * 3) = -3 + 12 - 9 = 0 (Perpendicular to v)
- (-3, 6, -3) ⋅ (4, 5, 6) = (-3 4) + (6 5) + (-3 * 6) = -12 + 30 - 18 = 0 (Perpendicular to w)
Finding a Perpendicular Vector to a Single Vector
While the cross product finds a vector perpendicular to two vectors, you might also want to find any vector perpendicular to a single given vector, say u = (u₁, u₂, u₃).
There are infinitely many vectors perpendicular to a single vector. A vector p = (p₁, p₂, p₃) is perpendicular to u if their dot product is zero:
u ⋅ p = u₁p₁ + u₂p₂ + u₃p₃ = 0
To find one such vector, you can pick arbitrary values for two components of p and solve for the third. For instance, if u = (1, 2, 3), you need 1p₁ + 2p₂ + 3*p₃ = 0. You could pick p₁=1, p₂=1, then 1 + 2 + 3p₃ = 0, so 3p₃ = -3, p₃ = -1. One possible perpendicular vector is (1, 1, -1).
However, the cross product provides a systematic way to get a unique direction (up to a scalar multiple) that is perpendicular to a pair of vectors, which is often what is needed in 3D problems.
