Question

If \(\vec{a}=2\hat{i}+3\hat{j}+4\hat{k}\) and \(\vec{b}=4\hat{i}-2\hat{j}+3\hat{k}\). Find \(|\vec{a}×\vec{b}|\).

(a) \(\sqrt{685}\)

(b) \(\sqrt{645}\)

(c) \(\sqrt{679}\)

(d) \(\sqrt{689}\)

The question was posed to me in an online quiz.

I want to ask this question from Product of Two Vectors-2 topic in division Vector Algebra of Mathematics – Class 12

Answer 1

Correct choice is (b) \(\sqrt{645}\)

To explain: Given that, \(\vec{a}=2\hat{i}+3\hat{j}+4\hat{k}\) and \(\vec{b}=4\hat{i}-2\hat{j}+3\hat{k}\)

∴ \(\vec{a}×\vec{b}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\2&3&4\\4&-2&3\end{vmatrix}\)

=\(\hat{i}(9—8)-\hat{j}(6-16)+\hat{k}(-4-12)\)

=\(17\hat{i}+10\hat{j}-16\hat{k}\)

∴\(|\vec{a}×\vec{b}|=\sqrt{17^2+10^2+(-16)^2}\)

=\(\sqrt{289+100+256}\)

=\(\sqrt{645}\)