Final answer:
A nonzero vector perpendicular to both vectors A and B is found by calculating the cross product of A and B, which results in the vector -5i + 2j - 7k.
Step-by-step explanation:
To find a nonzero vector that is perpendicular to both vectors A and B, we can use the cross product. The cross product of two vectors results in a third vector that is orthogonal to the plane containing the first two vectors. Given vector A = 2i + 9j - 4k, and vector B = i + j - k, we calculate their cross product (A x B).
Let's perform the cross product step-by-step:
- Write the components of vectors A and B into a matrix form, aligning i, j, and k components.
- Calculate the determinant of a 3x3 matrix that is formed by including unit vectors i, j, and k in the first row, components of vector A in the second row, and components of vector B in the third row.
- Apply the rule of the determinant to find the components of the resulting vector.
The resulting vector from A x B = (9(-1) - (-4)(1))i - (2(-1) - (-4)(1))j + ((2)(1) - (9)(1))k = (-9 + 4)i - (-2 + 4)j + (2 - 9)k = -5i + 2j - 7k.
So, the nonzero vector that is perpendicular to both vectors A and B is -5i + 2j - 7k.