Easy -1.2 This is a straightforward application of the section formula for dividing a line segment in a given ratio. Students need only to express position vectors and use the ratio 2:3 to find b, requiring basic vector addition and scalar multiplication with no problem-solving insight needed.
2.
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\section*{Figure 1}
Figure 1 shows a triangle \(O A C\) where \(O B\) divides \(A C\) in the ratio \(2 : 3\).
Show that \(\mathbf { b } = \frac { 1 } { 5 } ( 3 \mathbf { a } + 2 \mathbf { c } )\)
Show that \(\mathbf{b} = \frac{1}{5}(3\mathbf{a} + 2\mathbf{c})\)
(3)
M1 Attempts any two of \(\overrightarrow{AB} = \mathbf{b} - \mathbf{a}\), \(\overrightarrow{AC} = \mathbf{c} - \mathbf{a}\), \(\overrightarrow{BC} = \mathbf{c} - \mathbf{b}\); dM1 Uses given information; A1 Fully correct work including bracketing leading to given answer
**Part (a):**
Show that $\mathbf{b} = \frac{1}{5}(3\mathbf{a} + 2\mathbf{c})$ | (3) | M1 Attempts any two of $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$, $\overrightarrow{AC} = \mathbf{c} - \mathbf{a}$, $\overrightarrow{BC} = \mathbf{c} - \mathbf{b}$; dM1 Uses given information; A1 Fully correct work including bracketing leading to given answer
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