https://file.aiquickdraw.com/mathbot/file/1726736152437lvdo0h4d.jpg

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.

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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.

Tentukan komponen-komponen vektor v dengan titik awal P(x1, y1, z1) dan titik ujung Q(x2, y2, z2). Tentukan |v|. Gambarkan grafik v.

Learn how to determine the components of a vector v given points P and Q, calculate its magnitude, and sketch the vector. This problem involves key 3D vector concepts, suitable for high school students in grades 10-12.

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