Write $$ u $$ as the sum of two vectors, one parallel to $$ v $$ and the other perpendicular to $$ v $$. $$ u=3^{i}-4 j, v=-2 i+j $$

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To express the vector $$ u = 3\mathbf{i} - 4\mathbf{j} $$ as the sum of two vectors, one parallel to $$ v $$ and the other perpendicular to $$ v $$, we can follow these steps:

1. **Find the unit vector in the direction of $$ v $$**:
   $$
   v = -2\mathbf{i} + \mathbf{j}
   $$
   The magnitude of $$ v $$ is:
   $$
   \|v\| = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}
   $$
   The unit vector $$ \hat{v} $$ in the direction of $$ v $$ is:
   $$
   \hat{v} = \frac{v}{\|v\|} = \frac{-2\mathbf{i} + \mathbf{j}}{\sqrt{5}} = -\frac{2}{\sqrt{5}}\mathbf{i} + \frac{1}{\sqrt{5}}\mathbf{j}
   $$

2. **Project $$ u $$ onto $$ v $$**:
   The projection of $$ u $$ onto $$ v $$ is given by:
   $$
   \text{proj}_{v} u = \left( \frac{u \cdot v}{v \cdot v} \right) v
   $$
   First, we calculate $$ u \cdot v $$:
   $$
   u \cdot v = (3\mathbf{i} - 4\mathbf{j}) \cdot (-2\mathbf{i} + \mathbf{j}) = 3(-2) + (-4)(1) = -6 - 4 = -10
   $$
   Next, we calculate $$ v \cdot v $$:
   $$
   v \cdot v = (-2\mathbf{i} + \mathbf{j}) \cdot (-2\mathbf{i} + \mathbf{j}) = (-2)^2 + 1^2 = 4 + 1 = 5
   $$
   Now we can find the projection:
   $$
   \text{proj}_{v} u = \left( \frac{-10}{5} \right) v = -2v = -2(-2\mathbf{i} + \mathbf{j}) = 4\mathbf{i} - 2\mathbf{j}
   $$

3. **Find the vector perpendicular to $$ v $$**:
   The vector $$ w $$ that is perpendicular to $$ v $$ can be found by subtracting the projection from $$ u $$:
   $$
   w = u - \text{proj}_{v} u = (3\mathbf{i} - 4\mathbf{j}) - (4\mathbf{i} - 2\mathbf{j}) = (3 - 4)\mathbf{i} + (-4 + 2)\mathbf{j} = -\mathbf{i} - 2\mathbf{j}
   $$

4. **Combine the results**:
   We can now express $$ u $$ as the sum of the two vectors:
   $$
   u = \text{proj}_{v} u + w = (4\mathbf{i} - 2\mathbf{j}) + (-\mathbf{i} - 2\mathbf{j})
   $$

Thus, we have:
$$
u = (4\mathbf{i} - 2\mathbf{j}) + (-\mathbf{i} - 2\mathbf{j})
$$
This shows that $$ u $$ can be expressed as the sum of the vector $$ 4\mathbf{i} - 2\mathbf{j} $$ (parallel to $$ v $$) and the vector $$ -\mathbf{i} - 2\mathbf{j} $$ (perpendicular to $$ v $$).
$$ u = 4\mathbf{i} - 2\mathbf{j} + (-\mathbf{i} - 2\mathbf{j}) $$

Write as the sum of two vectors, one parallel to and the other
perpendicular to .

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