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It looks like you've uploaded an image with several vector-related problems. Let's go over them step by step. Here's a quick breakdown of the questions from the image:

1. **Problem 1 (Vector operations and plotting)**:
   - Vectors: \(\mathbf{a} = 2i - j\), \(\mathbf{b} = -2i + 3j\), \(\mathbf{c} = 4i + j\)
   - Required to calculate: \(\mathbf{a} + \mathbf{b}\), \(\mathbf{a} - \mathbf{c}\), \(2\mathbf{b} - \mathbf{a}\), and \(|\mathbf{a} + \mathbf{b}|\)
   - Plot vectors \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\), \(\mathbf{a} + \mathbf{b}\), \(\mathbf{a} - \mathbf{c}\) in the x, y plane.

2. **Problem 2 (Solving for unknowns)**:
   - Vectors: \(\mathbf{p} = 3i + uj\), \(\mathbf{q} = vi - 4j\), \(\mathbf{r} = 4i - 6j\)
   - Solve for \(u\) and \(v\) if \(\mathbf{p} - \mathbf{q} = \mathbf{r}\)
   - Find \(u\) if \(\mathbf{p}\) and \(\mathbf{r}\) are parallel.

3. **Problem 3 (Vector magnitudes and directions)**:
   - Given \(\mathbf{a} = -3i + 4j\)
   - Find a vector parallel to \(\mathbf{a}\) with magnitude 20.
   - Find a unit vector in the direction of \(\mathbf{a}\).

4. **Problem 4 (Component form of vectors)**:
   - Express the given vectors in the form \(\mathbf{a} = a_1i + a_2j\) for different magnitudes and directions from the diagrams.

5. **Problem 5 (Magnitude and direction of vectors)**:
   - Find magnitude and direction \(\theta\) of several vectors.

6. **Problem 6 (Vector operations in 3D)**:
   - Vectors: \(\mathbf{a} = 3i - j + 2k\), \(\mathbf{b} = 6i - 3j - 2k\), \(\mathbf{c} = i + j - 3k\)
   - Perform vector operations and find magnitudes, angles, and parallel vectors.

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Solve the problems related to vector algebra, including vector operations, plotting, solving for unknowns, vector magnitudes, unit vectors, and parallel vectors.

This page provides solutions to vector algebra problems involving vector addition, subtraction, magnitudes, and direction angles. It includes step-by-step explanations for calculating unit vectors, solving for unknown components, and determining parallelism. Suitable for university engineering students.

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