Questions: Solve for w where u=(0,-1,1,1) and v=(2,3,-1,0). w+2 v=-4 u

Transcript text: Solve for $\mathbf{w}$ where $\mathbf{u}=(0,-1,1,1)$ and $\mathbf{v}=(2,3,-1,0)$. \[ \mathbf{w}+2 \mathbf{v}=-4 \mathbf{u} \]

Solution

Solution Steps

To solve for $\mathbf{w}$, we need to isolate $\mathbf{w}$ in the given vector equation. We can do this by first calculating $2 \mathbf{v}$ and $-4 \mathbf{u}$, and then solving for $\mathbf{w}$ by subtracting $2 \mathbf{v}$ from $-4 \mathbf{u}$.

Step 1: Define the Vectors

We are given the vectors: \[ \mathbf{u} = \begin{bmatrix} 0 \\ -1 \\ 1 \\ 1 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 2 \\ 3 \\ -1 \\ 0 \end{bmatrix} \]

Step 2: Calculate $2\mathbf{v}$

We compute $2\mathbf{v}$: \[ 2\mathbf{v} = 2 \cdot \begin{bmatrix} 2 \\ 3 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 4 \\ 6 \\ -2 \\ 0 \end{bmatrix} \]

Step 3: Calculate $-4\mathbf{u}$

Next, we calculate $-4\mathbf{u}$: \[ -4\mathbf{u} = -4 \cdot \begin{bmatrix} 0 \\ -1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 4 \\ -4 \\ -4 \end{bmatrix} \]

Step 4: Solve for $\mathbf{w}$

Now, we can find $\mathbf{w}$ using the equation: \[ \mathbf{w} = -4\mathbf{u} - 2\mathbf{v} \] Substituting the values we calculated: \[ \mathbf{w} = \begin{bmatrix} 0 \\ 4 \\ -4 \\ -4 \end{bmatrix} - \begin{bmatrix} 4 \\ 6 \\ -2 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 - 4 \\ 4 - 6 \\ -4 + 2 \\ -4 - 0 \end{bmatrix} = \begin{bmatrix} -4 \\ -2 \\ -2 \\ -4 \end{bmatrix} \]