find a unit vector that is orthogonal to both i + j and i + k.

Gim Authors

2 min read

·

14-10-2024

43

0

Share

Finding a Unit Vector Orthogonal to Two Given Vectors

In linear algebra, the concept of orthogonality is fundamental. Two vectors are orthogonal if their dot product is zero. This means they are perpendicular to each other. Finding a vector orthogonal to two given vectors has various applications in fields like physics, engineering, and computer graphics.

Let's explore how to find a unit vector orthogonal to i + j and i + k, using insights from Stack Overflow.

Understanding the Problem

We are given two vectors:

• v1 = i + j
• v2 = i + k

Our goal is to find a unit vector u that is orthogonal to both v1 and v2. This means:

• u ⋅ v1 = 0
• u ⋅ v2 = 0

The Cross Product Solution

Stack Overflow user David G. Stork provides a concise solution: "The cross product of two vectors is always orthogonal to both of the original vectors."

This is a key insight. The cross product of v1 and v2 will yield a vector orthogonal to both.

Calculating the Cross Product

The cross product of two vectors in 3D space is defined as:

a x b = (a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)k

Where a = (a1, a2, a3) and b = (b1, b2, b3)

Applying this to our vectors:

• v1 = (1, 1, 0)
• v2 = (1, 0, 1)

We get:

• v1 x v2 = (11 - 00)i + (01 - 11)j + (10 - 11)k = i - j - k

Normalizing to a Unit Vector

The cross product i - j - k is orthogonal to both v1 and v2, but it's not a unit vector. To normalize it, we need to divide it by its magnitude.

The magnitude of a vector (a, b, c) is calculated as:

||(a, b, c)|| = √(a² + b² + c²) 

Therefore, the magnitude of i - j - k is:

||i - j - k|| = √(1² + (-1)² + (-1)²) = √3

Finally, the unit vector u orthogonal to both v1 and v2 is:

u = (i - j - k) / ||i - j - k|| = (1/√3)i - (1/√3)j - (1/√3)k

Verification

We can verify our answer by checking if the dot products of u with v1 and v2 are zero:

• u ⋅ v1 = ((1/√3) * 1) + ((-1/√3) * 1) + ((-1/√3) * 0) = 0
• u ⋅ v2 = ((1/√3) * 1) + ((-1/√3) * 0) + ((-1/√3) * 1) = 0

This confirms that u is indeed orthogonal to both v1 and v2.

Key Takeaways

• The cross product is a powerful tool for finding orthogonal vectors.
• Normalizing a vector makes it a unit vector, with a magnitude of 1.
• Verification ensures that the calculated vector meets the desired conditions.

Practical Applications

Finding orthogonal vectors is essential in various applications:

• Physics: Determining the direction of forces and magnetic fields.
• Engineering: Analyzing stress and strain in structures.
• Computer Graphics: Calculating surface normals for lighting and shading.

By understanding the concept of orthogonality and employing the cross product, we can solve problems and gain deeper insights in various fields.