Question

Let \( \vec{a} \) and \( \vec{b} \) be non-collinear vectors. If vector \( \vec{r} \) is coplanar with \( \vec{a} \) and \( \vec{b} \), then show that there exist unique scalars \( t_1 \) and \( t_2 \) such that \( \vec{r} = t_1 \vec{a} + t_2 \vec{b} \). For \( \vec{r} = 2\hat{i} + 7\hat{j} + 9\hat{k} \), \( \vec{a} = \hat{i} + 2\hat{j} \), \( \vec{b} = \hat{j} + 3\hat{k} \), find \( t_1, t_2 \).

Hint:

For coplanar vectors, express the given vector as a linear combination of the other two vectors. Solve the system of equations to find the scalars.

Answer 1

Step 1: Express the given vector equation.
We are given that \( \vec{r} = t_1 \vec{a} + t_2 \vec{b} \). So, the vector \( \vec{r} \) can be written as: \[ \vec{r} = t_1 (\hat{i} + 2\hat{j}) + t_2 (\hat{j} + 3\hat{k}) \] This simplifies to: \[ \vec{r} = t_1 \hat{i} + 2t_1 \hat{j} + t_2 \hat{j} + 3t_2 \hat{k} \] \[ \vec{r} = t_1 \hat{i} + (2t_1 + t_2) \hat{j} + 3t_2 \hat{k} \]

Step 2: Set the equation equal to the given vector \( \vec{r} = 2\hat{i} + 7\hat{j} + 9\hat{k} \).
We compare this with \( \vec{r} = 2\hat{i} + 7\hat{j} + 9\hat{k} \) to get the following system of equations: \[ t_1 = 2, 2t_1 + t_2 = 7, 3t_2 = 9 \]

Step 3: Solve the system of equations.
From \( 3t_2 = 9 \), we get: \[ t_2 = 3 \] Substitute \( t_2 = 3 \) into \( 2t_1 + t_2 = 7 \): \[ 2t_1 + 3 = 7 $\Rightarrow$ 2t_1 = 4 $\Rightarrow$ t_1 = 2 \]

Final Answer: The values of \( t_1 \) and \( t_2 \) are: \[ t_1 = 2, t_2 = 3 \]