A hyperplane is given by x+2y+3z=9 in which half space the points (1,2,3) and (0,1,-3) lie.

To determine in which half-space the points \((1, 2, 3)\) and \((0, 1, -3)\) lie with respect to the hyperplane given by the equation \(x + 2y + 3z = 9\), we can substitute the coordinates of each point into the equation of the hyperplane.

### Hyperplane Equation:
\[
x + 2y + 3z = 9
\]

### Step 1: Substituting the Point \((1, 2, 3)\)

\[
1 + 2(2) + 3(3) = 1 + 4 + 9 = 14
\]

\[
14 > 9
\]

So, the point \((1, 2, 3)\) lies in the positive half-space relative to the hyperplane.

### Step 2: Substituting the Point \((0, 1, -3)\)

\[
0 + 2(1) + 3(-3) = 0 + 2 - 9 = -7
\]

\[
-7 < 9
\]

So, the point \((0, 1, -3)\) lies in the negative half-space relative to the hyperplane.

### Conclusion:
- The point \((1, 2, 3)\) lies in the positive half-space.
- The point \((0, 1, -3)\) lies in the negative half-space.

Would you like further details or have any questions about this solution?

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Here are some related questions for further exploration:

1. How do you find the distance of a point from a hyperplane?
2. What is the geometric interpretation of a hyperplane in 3D space?
3. How do you determine the normal vector of a hyperplane given its equation?
4. What is the equation of a hyperplane in n-dimensional space?
5. How can you find the intersection of two or more hyperplanes?

**Tip:** When determining the position of a point relative to a hyperplane, always substitute the point's coordinates into the hyperplane equation to check the sign of the result.

Learn how to determine in which half-space points (1, 2, 3) and (0, 1, -3) lie with respect to the hyperplane given by the equation x + 2y + 3z = 9. Understand the concept of hyperplanes and half-spaces in linear algebra. Suitable for advanced high school students and beyond.

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