Given the vectors u=4i+6j-5k and v=-5i+3j-7k
a) Determine the vector c=a+b
b) Represent the vectors in the space.

Given the vectors \(\mathbf{u} = 4\mathbf{i} + 6\mathbf{j} - 5\mathbf{k}\) and \(\mathbf{v} = -5\mathbf{i} + 3\mathbf{j} - 7\mathbf{k}\), let’s proceed step by step:

### a) Determine the vector \(\mathbf{c} = \mathbf{u} + \mathbf{v}\)

The sum of two vectors is obtained by adding their respective components:

\[
\mathbf{c} = (4\mathbf{i} + 6\mathbf{j} - 5\mathbf{k}) + (-5\mathbf{i} + 3\mathbf{j} - 7\mathbf{k})
\]

Now, add the corresponding components:

- For \(\mathbf{i}\)-components: \(4 + (-5) = -1\)
- For \(\mathbf{j}\)-components: \(6 + 3 = 9\)
- For \(\mathbf{k}\)-components: \(-5 + (-7) = -12\)

Thus, the vector \(\mathbf{c}\) is:

\[
\mathbf{c} = -\mathbf{i} + 9\mathbf{j} - 12\mathbf{k}
\]

### b) Represent the vectors in space

To represent the vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{c}\) in 3D space, you would plot each vector starting from the origin (since they are position vectors). Here’s a breakdown of their coordinates:

1. **Vector \(\mathbf{u}\)** has coordinates \((4, 6, -5)\)
2. **Vector \(\mathbf{v}\)** has coordinates \((-5, 3, -7)\)
3. **Vector \(\mathbf{c}\)** has coordinates \((-1, 9, -12)\)

These vectors can be visualized in a 3D coordinate system with axes for \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) corresponding to the x, y, and z directions, respectively.

Would you like a visual representation of these vectors in space?

### Here are 5 related questions for further exploration:

1. How do you find the magnitude of vector \(\mathbf{c}\)?
2. How do you calculate the dot product of vectors \(\mathbf{u}\) and \(\mathbf{v}\)?
3. What is the cross product of \(\mathbf{u}\) and \(\mathbf{v}\)?
4. What is the angle between vectors \(\mathbf{u}\) and \(\mathbf{v}\)?
5. How would you normalize vector \(\mathbf{c}\)?

**Tip:** When adding vectors, always sum their components along each axis separately to ensure accuracy.

Given the vectors u=4i+6j-5k and v=-5i+3j-7k
a) Determine the vector c=u+v
b) Represent the vectors in the space.

Explore vector addition with u=4i+6j-5k and v=-5i+3j-7k. Learn how to calculate the resultant vector and visualize the vectors in 3D space. Ideal for Grades 10-12 students studying vector operations and 3D coordinate geometry.

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