Question

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

Short answer

\(\mathbf{z}\) is \(\langle 7, 0, -4 \rangle\).

Step-by-step solution

Subtract vector \(\mathbf{v}\) from \(\mathbf{u}\)

To subtract a vector from another, subtract corresponding components. \(\mathbf{u}\) – \(\mathbf{v}\) results in the new vector \(\langle 1 - 2, 2 - 2, 3 - (-1) \rangle = \langle -1, 0, 4 \rangle\)

Multiply vector \(\mathbf{w}\) by 2

This is scalar multiplication. Each component of the vector \(\mathbf{w}\) is to be multiplied by 2 resulting in the new vector \(2 \mathbf{w} = 2 \langle 4, 0, -4 \rangle = \langle 8, 0, -8 \rangle\)

Add the results

Now add up the two obtained vectors: \(\langle -1, 0, 4 \rangle\) + \(\langle 8, 0, -8 \rangle\) would result in the vector \(\langle -1 + 8, 0 + 0, 4 - 8 \rangle = \langle 7, 0, -4 \rangle\). This is \(\mathbf{z}\).