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The Two Adjacent Sides of a Parallelogram Are 2 ^ I − 4 ^ J + 5 ^ K and ^ I − 2 ^ J − 3 ^ K . \ Find the Unit Vector Parallel to One of Its Diagonals. Also, Find Its Area. - Mathematics

Sum

The two adjacent sides of a parallelogram are \[2 \hat{ i  } - 4 \hat{ j }  + 5 \hat{ k }  \text{ and }  \hat{ i } - 2 \hat{ j }  - 3\hat{ k }  .\]\  Find the unit vector parallel to one of its diagonals. Also, find its area. 

 
 
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Solution

\[\text{ Suppose }  \square ABCD \text{ is the given parallelogram and AC is its diagonal } . \]

\[\text{ Let } : \]

\[ \vec{AB} = 2 \hat{ i }  - 4 \hat{ j }  + 5 \hat{ k }  \]

\[ \vec{BC} = \hat{ i }  - 2 \hat{ j }  - 3 \hat{ k }  \]

\[ \therefore \text{ Diagonal }  \vec{AC} = \vec{AB} + \vec{BC} \]

\[ = 3 \hat{ i } - 6 \hat{ j }  + 2 \hat{ k }  \]

\[ \Rightarrow \left| \vec{AC} \right| = \sqrt{9 + 36 + 4}\]

\[ = 7\]

\[\text{ Unit vector parallel to } \vec{AC} =\frac{\vec{AC}}{\left| \vec{AC} \right|}\]

\[ =\frac{3 \hat{ i } - 6 \hat{ j }  + 2 \hat{ k } }{7}\]

\[\text{ Now } ,\]

\[ \vec{AB} \times \vec{BC} = \begin{vmatrix}\hat{ i }  & \hat{ j } & \hat{ k }  \\ 2 & - 4 & 5 \\ 1 & - 2 & - 3\end{vmatrix}\]

\[ = 22 \hat{ i }  + 11 \hat{ j }  + 0 \hat{ k } \]

\[ \Rightarrow \left| \vec{AB} \times \vec{AC} \right| = \sqrt{484 + 121}\]

\[ = \sqrt{605}\]

\[ = 11\sqrt{5}\]

\[Area of triangleABC=\frac{1}{2}\left| \vec{AB} \times \vec{AC} \right|\]

\[ = \frac{11\sqrt{5}}{2}\text{ sq . units } \]

 

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APPEARS IN

RD Sharma Class 12 Maths
Chapter 25 Vector or Cross Product
Exercise 25.1 | Q 31 | Page 31
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