(Solved): a) In the diagram below, \( M \) and \( N \) are midpoints of \( B C \) and \ ...

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a) In the diagram below, \( M \) and \( N \) are midpoints of \( B C \) and \( O C \) respectively. \( \overrightarrow{O B}=\underset{\sim}{p}, \overrightarrow{B C}=\underset{\sim}{q} \) and \( B N \) extends to \( D \) such that \( B D: B N=4: 3 \). i. Find the vector, \( \overrightarrow{O M} \) in terms of \( \underset{\sim}{p} \) and \( \underset{\sim}{q} \). (1 mark) ii. Find the vector, \( \overrightarrow{D C} \) in terms of \( \underset{\sim}{p} \) and \( \underset{\sim}{q} \). (4 marks) iii. Hence, show that \( O M \) is parallel to \( D C \) and that their lengths are in the ratio of \( 3: 2 \). \( (2 \) marks \( ) \) b) Determine whether the three vectors: \( \underset{\sim}{a}=3 \underset{\sim}{i}-4 \underset{\sim}{j}-2 \underset{\sim}{\underset{\sim}{j}} \underset{\sim}{b}=2 \underset{\sim}{i}-\underset{\sim}{j}+\underset{\sim}{k} \) and \( \underset{\sim}{c}=5 \underset{\sim}{i}-10 \underset{\sim}{j}-8 \underset{\sim}{k} \) are linearly dependent or independent. (3 marks)