The cross product of the vectors 3i + 4j – 5k and –i + j – 2k is,

To find the cross product of the vectors ( mathbf{a} = 3mathbf{i} + 4mathbf{j} – 5mathbf{k} ) and ( mathbf{b} = -mathbf{i} + mathbf{j} – 2mathbf{k} ), we can use the determinant of a matrix formed by the unit vectors and the components of the vectors.

The cross product ( mathbf{a} times mathbf{b} ) is given by the determinant:

[

mathbf{a} times mathbf{b} = begin{vmatrix}

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

3 & 4 & -5 \

-1 & 1 & -2

end{vmatrix}

]

Calculating the determinant, we have:

[

mathbf{a} times mathbf{b} = mathbf{i} begin{vmatrix} 4 & -5 \ 1 & -2 end{vmatrix} – mathbf{j} begin{vmatrix} 3 & -5 \ -1 & -2 end{vmatrix} + mathbf{k} begin{vmatrix} 3 & 4 \ -1 & 1 end{vmatrix}

]

Calculating each of the 2×2 determinants:

1. For ( mathbf