To find a vector parallel to $\mathbf{r} = \langle 3, -4 \rangle$ with a specific length, we need to follow these steps:

1. Find the magnitude of the given vector $\mathbf{r}$: $|\mathbf{r}| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

2. Unit vector in the direction of $\mathbf{r}$: A unit vector $\mathbf{u}$ in the same direction as $\mathbf{r}$ is obtained by dividing each component of $\mathbf{r}$ by its magnitude: $\mathbf{u} = \frac{\mathbf{r}}{|\mathbf{r}|} = \frac{\langle 3, -4 \rangle}{5} = \langle \frac{3}{5}, \frac{-4}{5} \rangle$ Now, $\mathbf{u} = \langle 0.6, -0.8 \rangle$.

i) Find a vector with length 10:

To get a vector of length 10, we multiply the unit vector by 10: $\mathbf{v}_1 = 10 \times \mathbf{u} = 10 \times \langle 0.6, -0.8 \rangle = \langle 6, -8 \rangle$ So, the vector of length 10 parallel to $\mathbf{r}$ is $\langle 6, -8 \rangle$.

ii) Find a vector with length 1:

To get a vector of length 1, we already have the unit vector $\mathbf{u}$, which is: $\mathbf{v}_2 = \langle 0.6, -0.8 \rangle$ So, the vector of length 1 parallel to $\mathbf{r}$ is $\langle 0.6, -0.8 \rangle$.

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Would you like more details on any of the steps, or have any further questions?

Here are 5 related questions to explore:

1. How do you find the magnitude of a 3D vector?
2. How can you verify that two vectors are parallel?
3. What is the dot product of two vectors and how is it related to angles?
4. How do you find the projection of one vector onto another?
5. How do you add and subtract vectors geometrically?

Tip: Remember that multiplying a unit vector by a scalar changes its length but not its direction!