Let the vectors $\mathbf{u} = i - 2j$ and $\mathbf{v} = 3i + 4j$. We will proceed step by step to compute the following:

(a) The Sum of the Vectors $\mathbf{u}$ and $\mathbf{v}$

To add two vectors, we add their corresponding components: $\mathbf{u} + \mathbf{v} = (i - 2j) + (3i + 4j)$ $\mathbf{u} + \mathbf{v} = (1i + 3i) + (-2j + 4j) = 4i + 2j$ So, the sum of the vectors is: $\boxed{\mathbf{u} + \mathbf{v} = 4i + 2j}$

(b) The Difference of the Vectors $\mathbf{u}$ and $\mathbf{v}$

To find the difference, subtract the components of $\mathbf{v}$ from the corresponding components of $\mathbf{u}$: $\mathbf{u} - \mathbf{v} = (i - 2j) - (3i + 4j)$ $\mathbf{u} - \mathbf{v} = (1i - 3i) + (-2j - 4j) = -2i - 6j$ So, the difference of the vectors is: $\boxed{\mathbf{u} - \mathbf{v} = -2i - 6j}$

(c) The Lengths (Magnitudes) of Vectors $\mathbf{u}$ and $\mathbf{v}$

To calculate the length of a vector $\mathbf{a} = ai + bj$, we use the formula: $|\mathbf{a}| = \sqrt{a^2 + b^2}$

• For $\mathbf{u} = i - 2j$, the length is: $|\mathbf{u}| = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}$

• For $\mathbf{v} = 3i + 4j$, the length is: $|\mathbf{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

So, the magnitudes are: $\boxed{|\mathbf{u}| = \sqrt{5}, \quad |\mathbf{v}| = 5}$

(d) The Unit Vectors in the Same Direction as $\mathbf{u}$ and $\mathbf{v}$

A unit vector is found by dividing a vector by its magnitude. For a vector $\mathbf{a}$, the unit vector in the same direction is $\hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}$.

• For $\mathbf{u} = i - 2j$, the unit vector is: $\hat{\mathbf{u}} = \frac{\mathbf{u}}{|\mathbf{u}|} = \frac{i - 2j}{\sqrt{5}} = \frac{1}{\sqrt{5}}i - \frac{2}{\sqrt{5}}j$
• For $\mathbf{v} = 3i + 4j$, the unit vector is: $\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|} = \frac{3i + 4j}{5} = \frac{3}{5}i + \frac{4}{5}j$

Thus, the unit vectors are: $\boxed{\hat{\mathbf{u}} = \frac{1}{\sqrt{5}}i - \frac{2}{\sqrt{5}}j, \quad \hat{\mathbf{v}} = \frac{3}{5}i + \frac{4}{5}j}$

Drawing Both the Original Vectors and the Computed Vectors

To draw these vectors, we plot them on a coordinate plane.

1. Original vectors $\mathbf{u}$ and $\mathbf{v}$:

   • $\mathbf{u}$ starts from the origin (0,0) and ends at (1, -2).
   • $\mathbf{v}$ starts from the origin (0,0) and ends at (3, 4).

2. Sum $\mathbf{u} + \mathbf{v}$: This vector starts from the origin and ends at (4, 2).

3. Difference $\mathbf{u} - \mathbf{v}$: This vector starts from the origin and ends at (-2, -6).

Would you like me to generate a visual illustration of the vectors for you?

---

Follow-Up Questions:

1. What is the angle between vectors $\mathbf{u}$ and $\mathbf{v}$?
2. How can we use the dot product to verify vector lengths?
3. What happens when you multiply $\mathbf{u}$ and $\mathbf{v}$ by scalars?
4. How do the unit vectors change if $\mathbf{u}$ and $\mathbf{v}$ are rotated by 90 degrees?
5. How does the vector cross product apply to $\mathbf{u}$ and $\mathbf{v}$ in 3D?

Tip:

Remember that vectors can be decomposed into components for easier manipulation, especially when combining them or finding angles between them.