How do you find unit vectors that are orthogonal to both i, j and i, k?

To find a unit vector that is orthogonal to both vectors i and j (which represent the x-axis and y-axis respectively in 3D space), as well as to both i and k (which represent the x-axis and z-axis), we can proceed as follows:

First, we can identify the vectors:

• i = (1, 0, 0)
• j = (0, 1, 0)
• k = (0, 0, 1)

To find a vector that is orthogonal to both i and j, we can use the cross product:

u = i × j = (1, 0, 0) × (0, 1, 0) = (0, 0, 1) = k

Next, to find a vector orthogonal to both i and k, we can again use the cross product:

v = i × k = (1, 0, 0) × (0, 0, 1) = (0, -1, 0) = -j

Both k and –j are orthogonal to the respective pairs of vectors. To find unit vectors from these results, we need to make sure their magnitudes equal 1.

The unit vector in the direction of k is:

u_{unit} = rac{1}{|k|} * k = (0, 0, 1)

The unit vector in the direction of –j is:

v_{unit} = rac{1}{|-j|} * (-j) = (0, -1, 0)

Thus, the unit vectors that are orthogonal to both pairs of vectors are:

• (0, 0, 1)
• (0, -1, 0)
• (0, 1, 0) (also orthogonal to both i, j and i, k)

Therefore, we conclude that the unit vectors orthogonal to both i, j and i, k can be expressed as (0, 0, 1) and (0, ±1, 0).