The task presented in the image asks for the following:

1. Determine the components of the vector v given the initial point $P(x_1, y_1, z_1)$ and the end point $Q(x_2, y_2, z_2)$.
2. Calculate the magnitude $|v|$ of the vector.
3. Sketch the graph of the vector.

For Problem 1:

The points are:

• $P(1, 0, 2)$
• $Q(3, 2, 1)$

1. Vector components:

The vector $\mathbf{v}$ from $P(x_1, y_1, z_1)$ to $Q(x_2, y_2, z_2)$ is calculated as: $\mathbf{v} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)$

Substitute the given points $P(1, 0, 2)$ and $Q(3, 2, 1)$:

$\mathbf{v} = (3 - 1, 2 - 0, 1 - 2) = (2, 2, -1)$

2. Magnitude of the vector:

The magnitude of $\mathbf{v}$, denoted $|\mathbf{v}|$, is calculated using the formula: $|\mathbf{v}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

For $\mathbf{v} = (2, 2, -1)$:

$|\mathbf{v}| = \sqrt{2^2 + 2^2 + (-1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3$

3. Graph of the vector:

The vector $\mathbf{v} = (2, 2, -1)$ originates from the point $P(1, 0, 2)$ and extends to $Q(3, 2, 1)$. This would be a 3D representation, showing the direction and magnitude of the vector.

Would you like to go through the next problems, such as problem 5 or 8, or perhaps explore something further?

Related questions:

1. How do we calculate vector components in 2D space?
2. What is the general formula for the magnitude of a vector in $n$-dimensional space?
3. Can you explain how to graph a vector in 2D versus 3D?
4. What is the geometric interpretation of a negative vector component?
5. How do parallel vectors relate in terms of their components?

Tip:

Always double-check the direction of your vector by verifying the points, as the vector's magnitude is dependent on the correct subtraction of coordinates.