The cross product of the vectors (2i -3j+ 4k) and (i + 4j - 5k) is

Given,

$\mathrm{A}=2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+4\hat{\mathrm{k}}, \mathrm{B}=\hat{\mathrm{i}}+4\hat{\mathrm{j}}-5\hat{\mathrm{k}}$

Substituting the value,

$\left|\begin{array}{ccc}\hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}}\\ 2& -3& 4\\ 1& 4& -5\end{array}\right|$

$=\hat{\mathrm{i}}[15-16]-\hat{\mathrm{j}}[-10-4]+\hat{\mathrm{k}}[8+3]=-\hat{\mathrm{i}}+14\hat{\mathrm{j}}+11\hat{\mathrm{k}}$

Hence the correct answer is $-\hat{\mathrm{i}}+14\hat{\mathrm{j}}+11\hat{\mathrm{k}}.$