Given OA = a and OB = b, express the following vectors in terms of a and b only: (i) AB (ii) OD. Also, express OE in two ways and find k and h.

Here's a step-by-step solution to the problem:

Part (a): Express the following vectors in terms of $\mathbf{a}$ and $\mathbf{b}$ only.

• a) (i) $\vec{AB}$ Given $\vec{OA} = \mathbf{a}$ and $\vec{OB} = \mathbf{b}$. Using the triangle law of vector addition:

  $$\vec{AB} = \vec{OB} - \vec{OA}$$ $$\vec{AB} = \mathbf{b} - \mathbf{a}$$

  The vector $\vec{AB}$ is $\boxed{\mathbf{b} - \mathbf{a}}$.

• a) (ii) $\vec{OD}$ First, find $\vec{OC}$. We are given $\vec{OB} = \frac{2}{5}\vec{OC}$. Rearranging this, we get:

  $$\vec{OC} = \frac{5}{2}\vec{OB}$$

  Since $\vec{OB} = \mathbf{b}$:

  $$\vec{OC} = \frac{5}{2}\mathbf{b}$$

  Next, D is the midpoint of AC. Therefore, $\vec{OD}$ can be expressed as the average of $\vec{OA}$ and $\vec{OC}$:

  $$\vec{OD} = \frac{1}{2}(\vec{OA} + \vec{OC})$$

  Substitute $\vec{OA} = \mathbf{a}$ and $\vec{OC} = \frac{5}{2}\mathbf{b}$:

  $$\vec{OD} = \frac{1}{2}\left(\mathbf{a} + \frac{5}{2}\mathbf{b}\right)$$ $$\vec{OD} = \frac{1}{2}\mathbf{a} + \frac{5}{4}\mathbf{b}$$

  The vector $\vec{OD}$ is $\boxed{\frac{1}{2}\mathbf{a} + \frac{5}{4}\mathbf{b}}$.

Part (b): Express $\vec{OE}$ in two different ways, hence find the values of $k$ and $h$.

• First expression for $\vec{OE}$ (using $\vec{AE} = k\vec{AB}$): We can express $\vec{OE}$ as $\vec{OA} + \vec{AE}$.

  $$\vec{OE} = \vec{OA} + \vec{AE}$$

  Substitute $\vec{OA} = \mathbf{a}$ and $\vec{AE} = k\vec{AB} = k(\mathbf{b} - \mathbf{a})$ (from part a(i)):

  $$\vec{OE} = \mathbf{a} + k(\mathbf{b} - \mathbf{a})$$ $$\vec{OE} = \mathbf{a} + k\mathbf{b} - k\mathbf{a}$$ $$\vec{OE} = (1 - k)\mathbf{a} + k\mathbf{b} \quad \text{(Equation 1)}$$

• Second expression for $\vec{OE}$ (using $\vec{OE} = h\vec{OD}$): We are given $\vec{OE} = h\vec{OD}$. Substitute $\vec{OD} = \frac{1}{2}\mathbf{a} + \frac{5}{4}\mathbf{b}$ (from part a(ii)):

  $$\vec{OE} = h\left(\frac{1}{2}\mathbf{a} + \frac{5}{4}\mathbf{b}\right)$$ $$\vec{OE} = \frac{h}{2}\mathbf{a} + \frac{5h}{4}\mathbf{b} \quad \text{(Equation 2)}$$

• Finding the values of $k$ and $h$: Equate the two expressions for $\vec{OE}$ (Equation 1 and Equation 2):

  $$(1 - k)\mathbf{a} + k\mathbf{b} = \frac{h}{2}\mathbf{a} + \frac{5h}{4}\mathbf{b}$$

  Since $\mathbf{a}$ and $\mathbf{b}$ are non-parallel vectors, their coefficients must be equal. Equating coefficients of $\mathbf{a}$:

  $$1 - k = \frac{h}{2} \quad \text{(Equation 3)}$$

  Equating coefficients of $\mathbf{b}$:

  $$k = \frac{5h}{4} \quad \text{(Equation 4)}$$

  Substitute Equation 4 into Equation 3:

  $$1 - \frac{5h}{4} = \frac{h}{2}$$

  Multiply the entire equation by 4 to eliminate denominators:

  $$4 - 5h = 2h$$ $$4 = 7h$$ $$h = \frac{4}{7}$$

  Substitute the value of $h$ back into Equation 4 to find $k$:

  $$k = \frac{5}{4} \times \frac{4}{7}$$ $$k = \frac{5}{7}$$

  The values are $\boxed{k = \frac{5}{7}}$ and $\boxed{h = \frac{4}{7}}$.

Part (c): State the ratio in which E divides:

• c) (i) AB We found $\vec{AE} = k\vec{AB}$ and $k = \frac{5}{7}$. So, $\vec{AE} = \frac{5}{7}\vec{AB}$. This means that E divides AB in the ratio $AE : EB$.

  $$AE : EB = k : (1 - k)$$ $$AE : EB = \frac{5}{7} : \left(1 - \frac{5}{7}\right)$$ $$AE : EB = \frac{5}{7} : \frac{2}{7}$$ $$AE : EB = 5 : 2$$

  E divides AB in the ratio $\boxed{5 : 2}$.

• c) (ii) OD We found $\vec{OE} = h\vec{OD}$ and $h = \frac{4}{7}$. So, $\vec{OE