How can I find two unit vectors that are orthogonal to both the vectors (3, 2, 1) and (1, 1, 0)?

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To find two unit vectors that are orthogonal to both vectors (3, 2, 1) and (1, 1, 0), we can follow these steps:

Step 1: Find the Cross Product

The first step is to find a vector that is orthogonal to both given vectors. This can be done using the cross product. For two vectors A = (3, 2, 1) and B = (1, 1, 0), the cross product A × B can be calculated as follows:

A × B = | i   j   k  |
            | 3   2   1  |
            | 1   1   0  |

Calculating this determinant gives:

A × B = i(2*0 - 1*1) - j(3*0 - 1*1) + k(3*1 - 2*1)

So:

A × B = i(0 - 1) - j(0 - 1) + k(3 - 2)
A × B = -i + j + k
A × B = (-1, 1, 1)

Step 2: Calculate the Magnitude of the Resulting Vector

Next, we need to find the magnitude of the vector (-1, 1, 1):

|V| = sqrt((-1)² + 1² + 1²)
|V| = sqrt(1 + 1 + 1)
|V| = sqrt(3)

Step 3: Normalize the Vector

To find a unit vector, we divide the vector by its magnitude:

U = V / |V|
U = (-1, 1, 1) / sqrt(3)
U = (-1/sqrt(3), 1/sqrt(3), 1/sqrt(3))

Step 4: Find the Second Unit Vector

To find a second unit vector that is also orthogonal to both original vectors, we can simply take the negative of the first unit vector:

U2 = -U
U2 = (1/sqrt(3), -1/sqrt(3), -1/sqrt(3))

Summary

The two unit vectors orthogonal to both (3, 2, 1) and (1, 1, 0) are:

  • U1 = (-1/sqrt(3), 1/sqrt(3), 1/sqrt(3))
  • U2 = (1/sqrt(3), -1/sqrt(3), -1/sqrt(3))

These unit vectors satisfy the condition of being orthogonal to the provided vectors.

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