Introduction:
To find a unit vector perpendicular to both vector A and vector B, we can use the cross product of the two vectors. The cross product of two vectors results in a vector that is perpendicular to both of the original vectors. To ensure that the resulting vector is a unit vector, we need to divide the resulting vector by its magnitude.

Step 1: Find the cross product:
To find the cross product of vector A and vector B, we can use the determinant method. The determinant of the cross product is calculated as follows:
```
i j k
2 1 0
1 -1 2
```
Expanding this determinant, we get:
```
= (1 * 2 - (-1 * 0))i - (2 * 2 - 1 * 0)j + (2 * (-1) - 1 * 1)k
= 2i - 4j + (-4)k
```
So the cross product of vector A and vector B is 2i - 4j - 4k.

Step 2: Normalize the resulting vector:
To convert the resulting vector into a unit vector, we need to divide it by its magnitude. The magnitude of a vector can be calculated using the formula:
```
|V| = √(Vx^2 + Vy^2 + Vz^2)
```
In this case, the magnitude of the resulting vector is:
```
|V| = √(2^2 + (-4)^2 + (-4)^2)
= √(4 + 16 + 16)
= √36
= 6
```
Dividing the resulting vector by its magnitude, we get:
```
(2/6)i + (-4/6)j + (-4/6)k
```
Simplifying further, we get:
```
(1/3)i - (2/3)j - (2/3)k
```
So the unit vector perpendicular to both vector A and vector B is (1/3)i - (2/3)j - (2/3)k.

Conclusion:
The unit vector perpendicular to both vector A=2i + j + k and vector B=i - j + 2k is (1/3)i - (2/3)j - (2/3)k.

Take their cross product and divide by products magnitude