How to Make a Vector Perpendicular to Another?

To make a vector perpendicular to another vector, you need to find a vector such that their dot product (also known as the scalar product) is exactly zero.

The relationship between the dot product of two vectors and the angle between them is fundamental to understanding perpendicularity. As the reference states:

The angle x between two vectors a and b can be found using the formula a⋅b = |a| |b| cos(x). For the vectors to be perpendicular (at right angles) then cos(x) = 0, so we know that the dot product or scalar product a⋅b must = 0. If you calculate the scalar product and show it = 0 the vectors must be perpendicular.

This means that the condition for two vectors to be perpendicular is simply that their dot product is zero. To "make" a vector perpendicular to a given vector, you must find components for the new vector that satisfy this zero-dot-product condition.

Finding a Perpendicular Vector

Let the original vector be v. We want to find a vector w such that v is perpendicular to w. Based on the definition above, this means we need to find w such that v ⋅ w = 0.

The method to find such a vector w depends on the dimension of the space you are working in (e.g., 2D or 3D). In any dimension greater than one, there are infinitely many vectors perpendicular to a given non-zero vector. Here's how to find one such vector:

In Two Dimensions (2D)

If your vector v is in 2D, let v = (v₁, v₂). We want to find w = (w₁, w₂) such that:

v ⋅ w = v₁w₁ + v₂w₂ = 0

A simple and common way to find a vector w that satisfies this equation is to swap the components of v and negate one of them.

• Method: Given v = (v₁, v₂), one vector perpendicular to v is w = (-v₂, v₁).
• Verification: Let's calculate the dot product:
  v ⋅ w = (v₁, v₂) ⋅ (-v₂, v₁) = v₁(-v₂) + v₂v₁ = -v₁v₂ + v₁v₂ = 0
  Since the dot product is 0, the vectors are perpendicular.
• Alternative: Another perpendicular vector is w = (v₂, -v₁).

Example (2D):

Let v = (3, 4).
Using the method, a perpendicular vector is w = (-4, 3).
Check: (3, 4) ⋅ (-4, 3) = (3)(-4) + (4)(3) = -12 + 12 = 0. They are perpendicular.

In Three Dimensions (3D)

If your vector v is in 3D, let v = (v₁, v₂, v₃). We want to find w = (w₁, w₂, w₃) such that:

v ⋅ w = v₁w₁ + v₂w₂ + v₃w₃ = 0

This equation represents a plane passing through the origin. Any vector w that lies in this plane is perpendicular to v. There are infinitely many solutions for w. To find one non-zero perpendicular vector (assuming v is not the zero vector):

• Method: Look at the components of v. If v is not the zero vector, at least one component is non-zero.
  1. If v₁ ≠ 0 or v₂ ≠ 0 (i.e., v is not purely along the z-axis), a simple perpendicular vector is w = (v₂, -v₁, 0).
     • Verification: v ⋅ w = (v₁, v₂, v₃) ⋅ (v₂, -v₁, 0) = v₁(v₂) + v₂(-v₁) + v₃(0) = v₁v₂ - v₁v₂ + 0 = 0. This works. If v₁ or v₂ is non-zero, w will be non-zero.
  2. If v₁ = 0 and v₂ = 0, then v must be (0, 0, v₃) with v₃ ≠ 0 (since v is non-zero). In this case, a simple perpendicular vector is w = (1, 0, 0) or w = (0, 1, 0).
     • Verification: v ⋅ w = (0, 0, v₃) ⋅ (1, 0, 0) = 0(1) + 0(0) + v₃(0) = 0. This works.

This method provides a straightforward way to construct a non-zero vector perpendicular to a given non-zero vector in 3D using the dot product principle.

Example (3D):

Let v = (1, 2, 3).
v₁=1, v₂=2. Since v₁ ≠ 0 or v₂ ≠ 0, use w = (v₂, -v₁, 0) = (2, -1, 0).
Check: v ⋅ w = (1, 2, 3) ⋅ (2, -1, 0) = (1)(2) + (2)(-1) + (3)(0) = 2 - 2 + 0 = 0. They are perpendicular.

Let v = (0, 0, 5).
Here v₁=0, v₂=0, v₃=5. Use w = (1, 0, 0).
Check: v ⋅ w = (0, 0, 5) ⋅ (1, 0, 0) = (0)(1) + (0)(0) + (5)(0) = 0. They are perpendicular.

In summary, making a vector perpendicular to another relies on ensuring their dot product is zero, and specific techniques derived from this condition allow you to construct such a vector in different dimensions.