All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

Equation of plane OABC: r = \lambdaa + \mub, for \lambda, \mu \in \mathbb{R}
Since c is on plane OABC, c = \lambdaa + \mub for \lambda, \mu are constants

Using the ratio theorem formula, \vec{ON}=\frac{4\mathbf{a}+3\mathbf{c}}{7}

Area ONC = Area OMC
\frac{1}{2} |{\frac{4\mathbf{a}+3\mathbf{c}}{7} \times \mathbf{c}}| = \frac{1}{2}|{\frac{1}{2}\mathbf{b} \times \mathbf{c}}|

\frac{1}{7} |{4 \mathbf{a} \times \mathbf{c} + 3\mathbf{c} \times \mathbf{c}}| = \frac{1}{2}|{\mathbf{b} \times (\lambda \mathbf{a} + \mu \mathbf{b})}|

\frac{2}{7} |{4\mathbf{a} \times (\lambda \mathbf{a} + \mu \mathbf{b})}| = |{\mathbf{b} \times \lambda \mathbf{a} + \mathbf{b} \times \mu \mathbf{b}}|

\frac{2}{7} |{4\mathbf{a} \times \lambda \mathbf{a} + 4\mathbf{a} \times \mu \mathbf{b}}| = \frac{1}{2}|{\mathbf{b} \times \lambda \mathbf{a}}|

\frac{8\mu}{7} |{\mathbf{a} \times \mathbf{b}}| = \lambda|{\mathbf{a} \times \mathbf{b}}|

\frac{8\mu}{7} = \lambda

KS Comments:

For (i), some students can rely on the law of parallelogram to show too. For (iii), students must know all their vector product manipulations well to do this.


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