Let's solve the given problem step-by-step.

Problem Breakdown

We are tasked with finding the position vector of $L$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

Information provided:

1. $A$ has a position vector $\mathbf{a}$, and $B$ has a position vector $\mathbf{b}$.
2. $K$ lies on line segment $AB$ such that $AK : KB = 1 : 2$.
3. $OALK$ forms a parallelogram.

Step 1: Find the position vector of $K$

Since $K$ divides $AB$ in the ratio $1:2$, we use the section formula for position vectors. The formula is:

$\mathbf{r}_K = \frac{m\mathbf{b} + n\mathbf{a}}{m+n},$

where $m:n = 2:1$. Substituting the values:

$\mathbf{r}_K = \frac{2\mathbf{b} + 1\mathbf{a}}{2+1} = \frac{2\mathbf{b} + \mathbf{a}}{3}.$

Thus, the position vector of $K$ is:

$\mathbf{r}_K = \frac{\mathbf{a} + 2\mathbf{b}}{3}.$

Step 2: Use the parallelogram property to find $L$

In a parallelogram, opposite sides are equal and parallel. This means:

$\mathbf{r}_L - \mathbf{r}_A = \mathbf{r}_K - \mathbf{r}_O.$

Rewriting this to express $\mathbf{r}_L$:

$\mathbf{r}_L = \mathbf{r}_A + (\mathbf{r}_K - \mathbf{r}_O).$

Substitute the given vectors:

• $\mathbf{r}_A = \mathbf{a}$,
• $\mathbf{r}_K = \frac{\mathbf{a} + 2\mathbf{b}}{3}$,
• $\mathbf{r}_O = \mathbf{0}$ (since $O$ is the origin).

$\mathbf{r}_L = \mathbf{a} + \left(\frac{\mathbf{a} + 2\mathbf{b}}{3} - \mathbf{0}\right).$

Simplify:

$\mathbf{r}_L = \mathbf{a} + \frac{\mathbf{a} + 2\mathbf{b}}{3}.$

Combine terms:

$\mathbf{r}_L = \frac{3\mathbf{a}}{3} + \frac{\mathbf{a} + 2\mathbf{b}}{3}.$

$\mathbf{r}_L = \frac{3\mathbf{a} + \mathbf{a} + 2\mathbf{b}}{3}.$

$\mathbf{r}_L = \frac{4\mathbf{a} + 2\mathbf{b}}{3}.$

Thus, the position vector of $L$ is:

$\mathbf{r}_L = \frac{4\mathbf{a} + 2\mathbf{b}}{3}.$

Final Answer:

$\mathbf{r}_L = \frac{4\mathbf{a} + 2\mathbf{b}}{3}.$

Would you like further details or explanations on any part of this solution?

Related Questions:

1. What is the general formula for dividing a line segment in a given ratio in terms of vectors?
2. How can you verify the parallelogram property using vector addition?
3. If $K$ divides $AB$ in a different ratio, how would that affect the position vector of $L$?
4. What would the position vector of $L$ be if $K$ was the midpoint of $AB$?
5. Can you derive a general method to find the position vector of points in a parallelogram?

Tip:

When working with vectors and ratios, always use the section formula carefully, ensuring correct placement of the weights.