Question

find unit vector (2i^+3j^+4k) × (5i^+ 6j^+7k^)​

Answer 1

GIVEN :–

$\\ \bf \: \implies \: (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) \times (5 \hat{i} + 6 \hat{j} + 7 \hat{k})$

TO FIND :–

• Unit vector = ?

SOLUTION :–

$\\ \bf \: \vec{A}= \: (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) \times (5 \hat{i} + 6 \hat{j} + 7 \hat{k})$

• We should write this as –

$\\ \bf \: \vec{A}= \left|\begin{array}{ccc}\bf \hat{i} & \bf \hat{j} & \bf\hat{k} \\ \\ \bf2 & \bf3& \bf4 \\ \\ \bf5& \bf6& \bf7\end{array}\right|$

$\\ \bf \: \vec{A}= \hat{i}\left|\begin{array}{cc}\bf \bf3& \bf4 \\ \\ \bf6& \bf7\end{array}\right| - \hat{j}\left|\begin{array}{cc}\bf \bf2& \bf4 \\ \\ \bf5& \bf7\end{array}\right| + \hat{k}\left|\begin{array}{cc}\bf \bf2& \bf3 \\ \\ \bf5& \bf6\end{array}\right|$

$\\ \bf \: \vec{A}= \hat{i}(21 - 24)- \hat{j}(14 - 20) + \hat{k}(12 - 15)$

$\\ \bf \: \vec{A}= \hat{i}( - 3)- \hat{j}( - 6) + \hat{k}( - 3)$

$\\ \bf \: \vec{A}= - 3 \hat{i} + 6 \hat{j} - 3 \hat{k}$

• Unit vector –

$\\ \implies\bf \: \hat{A}= \dfrac{ \vec{A}}{ |A| }$

• Here –

$\\ \:\: {\huge{.}}\bf \:\: \hat{A}= Unit\:\: vector$

$\\ \:\: {\huge{.}}\bf \:\: \vec{A}= vector$

$\\ \:\: {\huge{.}}\bf \:\:|A|= Magnitude$

$\\ \implies \bf \hat{A}= \dfrac{ - 3 \hat{i} + 6 \hat{j} - 3 \hat{k}}{ \sqrt{ {( - 3)}^{2} + {(6)}^{2} + {( - 3)}^{2} } }$

$\\ \implies \bf \hat{A}= \dfrac{ - 3 \hat{i} + 6 \hat{j} - 3 \hat{k}}{ \sqrt{9 + 36 + 9} }$

$\\ \implies \bf \hat{A}= \dfrac{ - 3 \hat{i} + 6 \hat{j} - 3 \hat{k}}{ \sqrt{54} }$

$\\ \implies \bf \hat{A}= \dfrac{ - 3 \hat{i} + 6 \hat{j} - 3 \hat{k}}{ 3\sqrt{6} }$

$\\ \implies \large{ \boxed{ \bf \hat{A}= \dfrac{ - \hat{i} + 2 \hat{j} - \hat{k}}{\sqrt{6}}} }$