Express `-hat(i)-3hat(j)+4hat(k)` as the linear combination of the vectors `2hat(i)+hat(j)-4hat(k)`, `2hat(i)-hat(j)+3hat(k)` and `3hat(i)+hat(j)-2hat(k)`.

Explanation

To express the vector \(-\hat{i} - 3\hat{j} + 4\hat{k}\) as a linear combination of the vectors \(\mathbf{a} = 2\hat{i} + \hat{j} - 4\hat{k}\), \(\mathbf{b} = 2\hat{i} - \hat{j} + 3\hat{k}\), and \(\mathbf{c} = 3\hat{i} + \hat{j} - 2\hat{k}\), we can follow these steps:

### Step 1: Set up the equation
We want to express \(-\hat{i} - 3\hat{j} + 4\hat{k}\) in terms of \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\):
\[ -\hat{i} - 3\hat{j} + 4\hat{k} = x(2\hat{i} + \hat{j} - 4\hat{k}) + y(2\hat{i} - \hat{j} + 3\hat{k}) + z(3\hat{i} + \hat{j} - 2\hat{k}) \]

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