To find a unit vector that is orthogonal to both i and j as well as i and k, we first need to understand what it means for vectors to be orthogonal. Two vectors are orthogonal if their dot product is zero.
The standard basis vectors in three-dimensional space are:
- i = (1, 0, 0)
- j = (0, 1, 0)
- k = (0, 0, 1)
We can find a vector that is orthogonal to both i and j using the cross product. The cross product of i and j is:
i x j = | i j k |
| 1 0 0 |
| 0 1 0 |Calculating this determinant, we get:
i x j = k = (0, 0, 1)Now, we need to find a unit vector that is orthogonal to both i and k. Again, we use the cross product:
i x k = | i j k |
| 1 0 0 |
| 0 0 1 |This results in:
i x k = -j = (0, -1, 0)So we have two perpendicular vectors: (0, 0, 1) and (0, -1, 0). To create a unit vector from either of these, we can normalize them. A unit vector is created by dividing the vector by its magnitude.
The magnitude of (0, 0, 1) is 1, and the magnitude of (0, -1, 0) is also 1:
- Unit vector in the direction of k: (0, 0, 1)
- Unit vector in the direction of -j: (0, -1, 0)
Thus, both (0, 0, 1) and (0, -1, 0) are unit vectors that are orthogonal to the specified vectors. You can choose either depending on the direction you want.