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The image provides some mathematical content related to vectors, including their length (or magnitude), dot product, and the concept of quadrance. I'll break down the content step by step:

### 1. **Length (Magnitude) of a Vector**
The length of a vector \( \mathbf{v} = \begin{pmatrix} a \\ b \\ c \end{pmatrix} \) is defined as:
\[
|\mathbf{v}| = \sqrt{Q(\mathbf{v})} = \sqrt{a^2 + b^2 + c^2}
\]
Where \( Q(\mathbf{v}) \) represents the "quadrance" or the square of the magnitude.

In the example from the image, a vector \( \mathbf{w} = \begin{pmatrix} -4 \\ 6 \\ 4 \end{pmatrix} \) has a magnitude:
\[
|\mathbf{w}| = \sqrt{(-4)^2 + 6^2 + 4^2} = \sqrt{16 + 36 + 16} = \sqrt{68}
\]
So, the length of \( \mathbf{w} \) is \( \sqrt{68} \).

### 2. **Distance Between Two Points**
The distance between two points \( \mathbf{C} = [0, 2, 1] \) and \( D = [4, 2, 3] \) can be calculated by the formula:
\[
\text{Distance} = \sqrt{(4 - 0)^2 + (2 - 2)^2 + (3 - 1)^2} = \sqrt{4^2 + 0^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20}
\]
So, the distance is \( \sqrt{20} \).

### 3. **Dot Product**
For two vectors \( \mathbf{v_1} = \begin{pmatrix} a_1 \\ b_1 \\ c_1 \end{pmatrix} \) and \( \mathbf{v_2} = \begin{pmatrix} a_2 \\ b_2 \\ c_2 \end{pmatrix} \), the dot product is defined as:
\[
\mathbf{v_1} \cdot \mathbf{v_2} = a_1 a_2 + b_1 b_2 + c_1 c_2
\]
In the given example:
\[
\mathbf{v_1} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \quad \mathbf{v_2} = \begin{pmatrix} -1 \\ 5 \\ 3 \end{pmatrix}
\]
The dot product is:
\[
1 \times (-1) + 2 \times 5 + 3 \times 3 = -1 + 10 + 9 = 18
\]

### 4. **Observation About the Quadrance**
The quadrance \( Q(\mathbf{v}) \) of a vector is related to the dot product. In fact, the quadrance is:
\[
Q(\mathbf{v}) = \mathbf{v} \cdot \mathbf{v} = a^2 + b^2 + c^2
\]
Thus, the dot product of a vector with itself yields its quadrance.

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### Related Questions:
1. How do you compute the length of a vector in n-dimensional space?
2. What is the geometrical meaning of the dot product?
3. Can the dot product of two vectors be negative? What does that mean geometrically?
4. How is the concept of quadrance different from the squared distance between two points?
5. What is the relationship between the angle between two vectors and their dot product?

### Tip:
The dot product can help determine if two vectors are perpendicular: if their dot product equals zero, the vectors are orthogonal.

Find the length of a vector, the distance between two points, and compute the dot product of two vectors.

Learn how to compute the length of a vector, find the distance between two points, and calculate the dot product of two vectors. This problem covers key concepts in vector mathematics, including vector length, distance formula, and dot product calculation. Suitable for high school students in Grades 10-12.

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