How can I find a unit vector that is orthogonal to both the vectors i + j and i + k?

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To find a unit vector that is orthogonal to both vectors i + j and i + k, we can follow a systematic approach using the cross product.

The vectors can be represented as:

  • v1 = i + j = (1, 1, 0)
  • v2 = i + k = (1, 0, 1)

Now, we will calculate the cross product of v1 and v2:


v1 = (1, 1, 0)
v2 = (1, 0, 1)

cross product: v1 x v2 = |i    j    k|
                      |1    1    0|
                      |1    0    1|

Using the determinant method to calculate this, we get:


v1 x v2 = i(1*1 - 0*0) - j(1*1 - 1*1) + k(1*0 - 1*1)
       = i(1) - j(1 - 0) + k(0 - 1)
       = (1, -1, -1)

So, the vector orthogonal to both i + j and i + k is:


v = (1, -1, -1)

Now to find the unit vector, we need to normalize this vector. The formula for the magnitude of a vector v = (x, y, z) is:


magnitude = √(x² + y² + z²)

Calculating the magnitude:


magnitude = √(1² + (-1)² + (-1)²)
          = √(1 + 1 + 1)
          = √3

Now, to get the unit vector u that is in the same direction as our orthogonal vector v, we divide each component of v by the magnitude:


u = (1/√3, -1/√3, -1/√3)

Thus, the unit vector orthogonal to both i + j and i + k is:


u = (1/√3, -1/√3, -1/√3)

This unit vector not only maintains the property of orthogonality but is also normalized to a length of 1, fulfilling the requirement of being a unit vector.

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