Solving for λ and μ in Vector Equation λr + μs = t

Since you haven't provided the specific vectors $r, s,$ and $t$, I will demonstrate the method using a general approach. To solve for scalars $\lambda$ and $\mu$ in a vector equation of the form $\lambda \vec{r} + \mu \vec{s} = \vec{t}$, we use the property of linear independence in 2D or 3D space.

Answer

To find the constants $\lambda$ and $\mu$, express the equation as a system of linear equations by equating the components (i and j, or x and y) on both sides. Solve the resulting system using substitution or elimination methods.

Explanation

1. Set up the vector equation Write the vectors $\vec{r}, \vec{s},$ and $\vec{t}$ in component form. If $\vec{r} = \begin{pmatrix} r_1 \\ r_2 \end{pmatrix}$, $\vec{s} = \begin{pmatrix} s_1 \\ s_2 \end{pmatrix}$, and $\vec{t} = \begin{pmatrix} t_1 \\ t_2 \end{pmatrix}$, the equation becomes: $\lambda \begin{pmatrix} r_1 \\ r_2 \end{pmatrix} + \mu \begin{pmatrix} s_1 \\ s_2 \end{pmatrix} = \begin{pmatrix} t_1 \\ t_2 \end{pmatrix}$ We multiply the scalar constants by the respective vector components.

2. Form a system of linear equations Extract the individual equations by equating the horizontal ($x$) components and vertical ($y$) components: $1) \quad \lambda r_1 + \mu s_1 = t_1$ $2) \quad \lambda r_2 + \mu s_2 = t_2$ This creates two equations with two unknowns, which is solvable for any non-parallel vectors.

3. Solve for the constants Use either elimination or substitution. For instance, multiply equation (1) by $r_2$ and equation (2) by $r_1$ to align the $\lambda$ terms, then subtract them to eliminate $\lambda$ and solve for $\mu$. Once $\mu$ is found, substitute it back into either equation to find $\lambda$. Solving the system isolates the values of $\lambda$ and $\mu$ that satisfy the linear combination.

Final Answer

The values of $\lambda$ and $\mu$ are found by solving the simultaneous system derived from the vector components: $\boxed{\lambda = \dots, \mu = \dots}$

Common Mistakes

• Component Mixing: Students often add or subtract vectors incorrectly by accidentally combining the $x$-component of one vector with the $y$-component of another. Always write them out in columns to stay organized.
• Sign Errors: Forgetting to distribute the negative sign when multiplying by a negative scalar or when performing elimination/subtraction. Always distribute the scalar to both components of the vector.

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Tutor Note: If you provide the specific vectors $\vec{r}, \vec{s},$ and $\vec{t}$ from your worksheet, I can calculate the exact numerical values for you immediately!