Question

if \[\vec{a} = \hat{ i }- 2\hat{ j }  + 3 \hat{ k }  , \text{ and }  \vec{b} = 2 \hat{ i }  + 3 \hat{ j }  - 5 \hat{ k }  ,\]  then find \[\vec{a} \times \vec{b} .\]  Verify th at \[\vec{a} \text{ and }  \vec{a} \times \vec{b}\] are perpendicular to each other.

 

 

 

 

Answer 1

\[\text{ Given } : \]
\[ \vec{a} = \stackrel\frown {i }  - 2 \stackrel\frown {j } + 3 \stackrel\frown {k } \]
\[ \vec{b} = 2 \stackrel\frown{ i } + 3 \stackrel\frown{ j }   - 5 \stackrel\frown {k } \]

\[\vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j }  & \hat{ k }  \\ 1 & - 2 & 3 \\ 2 & 3 & - 5\end{vmatrix}\]
\[ = \hat{ i }  + 11 \hat{ j }  + 7 \hat{ k }  \]
\[\text{ Now } ,\]
\[ \vec{a} . \left( \vec{a} \times \vec{b} \right) = 1 - 22 + 21\]
\[ = 0\]
\[\text{ Thus, } \vec{a} \text{ is perpendicular to } \vec{a} \times \vec{b} .\]