What is a unit vector perpendicular to the plane determined by the vectors A = 2i + 4j and B = i + j - k?

To find a unit vector that is perpendicular to the plane defined by the vectors A and B, we first need to calculate the cross product of these two vectors. The cross product will yield a vector that is orthogonal to both A and B.

Given:

A = 2i + 4j,

B = i + j - k.

The cross product ( A \times B ) can be computed using the determinant of a matrix constructed from the unit vectors i, j, k and the components of A and B:

[

A \times B = \begin{vmatrix}

\mathbf{i} & \mathbf{j} & \mathbf{k} \

2 & 4 & 0 \

1 & 1 & -1

\end{vmatrix}

]

Calculating this determinant, we can break it down:

Expand along the first row:

• The coefficient of ( \mathbf{i} ) is ( (4 \cdot -1 - 0 \cdot 1) = -4 ).

• The coefficient of ( \mathbf{j} ) is ( -(2 \cdot -1 - 0 \cdot 1) = 2 ).

• The