Consider a vector 𝑣=(𝑥−7,2,1)∈𝑅3*v*=(*x*−7,2,1)∈R3. Find the value of 𝑥*x* so that the length of the vector 𝑣*v* is minimum.


To minimize the length of the vector \(\mathbf{v} = (x - 7, 2, 1)\), we need to minimize its Euclidean norm (or magnitude).

The Euclidean norm of a vector \(\mathbf{v} = (a, b, c)\) in \(\mathbb{R}^3\) is given by:

\[
\|\mathbf{v}\| = \sqrt{a^2 + b^2 + c^2}
\]

For the vector \(\mathbf{v} = (x - 7, 2, 1)\), the norm is:

\[
\|\mathbf{v}\| = \sqrt{(x - 7)^2 + 2^2 + 1^2}
\]

Simplifying inside the square root:

\[
\|\mathbf{v}\| = \sqrt{(x - 7)^2 + 4 + 1} = \sqrt{(x - 7)^2 + 5}
\]

To minimize \(\|\mathbf{v}\|\), we need to minimize the expression under the square root, which is \((x - 7)^2 + 5\). 

Since \(5\) is a constant, we need to minimize \((x - 7)^2\). The minimum value of \((x - 7)^2\) occurs when \((x - 7) = 0\), because the square of any real number is always non-negative, and it achieves its minimum value of 0 when the number itself is 0.

Thus:

\[
x - 7 = 0 \implies x = 7
\]

Therefore, the value of \(x\) that minimizes the length of the vector \(\mathbf{v}\) is:

\[
\boxed{7}
\]

Would you like any more details or have any questions?

Here are 5 questions you might want to ask next:
1. How do you calculate the distance between two points in \(\mathbb{R}^3\)?
2. What is the significance of a vector’s direction and magnitude?
3. How do you determine if two vectors are orthogonal?
4. How can you find the projection of one vector onto another?
5. What is the importance of unit vectors in vector operations?

**Tip:** When minimizing or maximizing functions involving squared terms, always consider the point where the squared term itself is zero, as it often simplifies the problem significantly.

Learn how to minimize the length of a vector v = (x - 7, 2, 1) in R^3 by finding the value of x. This problem involves the Euclidean norm of vectors and its minimization. Suitable for advanced high school students or those studying vector algebra.

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