Working with Standard Basis Vectors

1. If \(\mathbf{v} = 2 \mathbf{i} - 3 \mathbf{j}\) and \(\mathbf{w} = -\mathbf{i} + 2 \mathbf{j}\), find \(4\mathbf{v}-\mathbf{w}.\)
2. If \(\mathbf{v} = 2\mathbf{i} + 3 \mathbf{j} -\mathbf{k}\) and \(\mathbf{w} = -\mathbf{i}-2\mathbf{j}+4\mathbf{k}\), find \(2 \mathbf{v}-3 \mathbf{w}.\)

Solution

(a) \[\begin{align*} 4\mathbf{v} - \mathbf{w} = 4 (2 \mathbf{i} - 3 \mathbf{j}) - (-\mathbf{i} + 2 \mathbf{j}) = (8 \mathbf{i} - 12 \mathbf{j}) + (\mathbf{i} - 2 \mathbf{j}) = 9 \mathbf{i} - 14 \mathbf{j} \end{align*}\]

(b) \[\begin{align*} 2 \mathbf{v} - 3 \mathbf{w} &= 2(2 \mathbf{i} + 3 \mathbf{j} - \mathbf{k}) - 3(-\mathbf{i} - 2 \mathbf{j} + 4 \mathbf{k}) \\ & =(4 \mathbf{i} + 6 \mathbf{j} - 2 \mathbf{k}) + (3 \mathbf{i} + 6 \mathbf{j} - 12 \mathbf{k}) = 7 \mathbf{i} + 12 \mathbf{j} - 14 \mathbf{k} \end{align*}\]