Question

Find the vector \(z,\) given that \(\mathbf{u}=\langle 1,2,3\rangle\) \(\mathbf{v}=\langle 2,2,-1\rangle,\) and \(\mathbf{w}=\langle 4,0,-4\rangle\) \(2 \mathbf{u}+\mathbf{v}-\mathbf{w}+3 \mathbf{z}=\mathbf{0}\)

Short answer

\(\mathbf{z} = -\langle 0,2,2 \rangle\)

Step-by-step solution

Write down the vector equation

The vector equation we want to solve is \(2 \mathbf{u}+\mathbf{v}-\mathbf{w}+3 \mathbf{z}=\mathbf{0}\).

Plug in the known vectors

The vectors \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \) are given by \( \mathbf{u}=\langle 1,2,3\rangle\), \( \mathbf{v}=\langle 2,2,-1\rangle\), and \( \mathbf{w}=\langle 4,0,-4\rangle\). If we insert these into the equation we get \(2 \langle 1,2,3\rangle + \langle 2,2,-1\rangle - \langle 4,0,-4\rangle + 3 \mathbf{z} = \langle 0,0,0\rangle\).

Perform the addition and subtraction

The result of the addition and subtraction of the known vectors is \( \langle 0,6,6 \rangle + 3 \mathbf{z} = \langle 0,0,0\rangle \).

Solve the equation for vector \( z \)

When we bring \( \langle 0,6,6 \rangle \) on the other side of the equals sign, the equation becomes \( 3 \mathbf{z} = -\langle 0,6,6 \rangle \). Dividing both sides by 3 gives the solution for \( \mathbf{z} \), which is \( \mathbf{z} = -\langle 0,2,2 \rangle \).