Let u = i − 2j and v = 3i+4j be vectors. Compute the following for vectors u and v: (a) (d) the sum (b) the difference the unit vectors in the same direction and their lengths (c) the lengths and draw both the original vectors and the vectors computed from them in the same figure

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?

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### 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.

Let u = i − 2j and v = 3i + 4j be vectors. Compute the following for vectors u and v: (a) the sum (b) the difference (c) the lengths (d) the unit vectors in the same direction and their lengths, and draw both the original vectors and the vectors computed from them in the same figure.

Explore vector algebra with a step-by-step solution for the vectors u = i - 2j and v = 3i + 4j. Learn how to compute the sum, difference, magnitudes, and unit vectors, and visualize these vectors on a coordinate plane. Suitable for high school students (Grades 10-12).

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