**Finding a Vector Perpendicular to a and b**

To find a vector p that is perpendicular to both a and b, we can use the cross product of a and b. The cross product of two vectors produces a vector that is perpendicular to both of them.

The cross product of two vectors a and b is given by the formula:

p = a x b

where p is the vector perpendicular to both a and b.

Given vectors a = 4i + 5j - k and b = i - 4j + 5k, we can calculate the cross product.

**Calculating the Cross Product**

To calculate the cross product, we need to determine the components of p using the determinant method:

i j k

4 5 -1

1 -4 5

Using the determinant method, we can calculate the cross product as follows:

p = (5 * 5 - (-4 * -1))i - ((4 * 5 - (-1 * 1))j + (4 * (-4) - (1 * 5))k

Simplifying the equation, we get:

p = 21i + 9j - 21k

Therefore, the vector p that is perpendicular to both a and b is given by p = 21i + 9j - 21k.

**Calculating p•q**

To calculate the dot product of p and q, we need to find the dot product of their respective components.

p•q = (21 * 3) + (9 * 0) + (-21 * 1)

Simplifying the equation, we get:

p•q = 63 - 21

p•q = 42

Therefore, p•q = 42.

**Summary**

To find a vector p that is perpendicular to both a = 4i + 5j - k and b = i - 4j + 5k, we calculated the cross product of a and b using the determinant method. The resulting vector p = 21i + 9j - 21k is perpendicular to both a and b.

We then calculated the dot product of p and q = 3i + j + k to find p•q. The dot product of p and q is equal to 42.

Hence, the vector p that is perpendicular to both a and b and satisfies p•q = 21 is p = 21i + 9j - 21k.