Simple Hyperplane Equation - What am I Missing?

• cowmoo32
In summary, the conversation is about finding the equation of a hyperplane in R4 that passes through a given point and is normal to a given vector. The equation is solved by substituting the point into the equation and obtaining the value for k. However, there seems to be an error in the given values as the final answer obtained does not match the expected answer. Further clarification is needed to find the mistake.
cowmoo32
I'm trying to learn linear algebra using Schaum's outline and this is one of the practice problems:

Find and equation of the hyperplane H in R4 that passes through P(3,-4,1,-2) and is normal to u=[2,5,-6,-3]

The equation is in the form 2x1+5x2-6x3-3x4 = k

They say to substitute P into the equation to obtain k=-26 but I keep coming up with k=-14. This is simple math but I'm clearly missing something. Where am I going wrong?

It's probably a typo.

Let X be a point in the hyperplane. Then u.(X-P)=0 should be your equation.

So, 2(x1 - 3) + 5(x2 + 4) - 6(x3 - 1) - 3(x4 + 2) = 0

2x1 + 5x2 - 6x3 - 3x4 + 14 = 0

Same answer you get. Is there a typo somewhere?

There has to be a typo because the next problem is similar and the answer there is correct.

It seems like you have the right idea, but there may be a small mistake in your calculations. Let's go through the steps to double check and see where the discrepancy may be coming from.

First, let's recall the general equation of a hyperplane in R4 is of the form ax1 + bx2 + cx3 + dx4 = k, where a, b, c, and d are the coefficients and k is a constant.

Since we know that the hyperplane in question is normal to u = [2, 5, -6, -3], we can use this vector as the normal vector and plug it into the equation. This gives us the equation 2x1 + 5x2 - 6x3 - 3x4 = k.

Next, we can substitute the coordinates of point P (3, -4, 1, -2) into this equation to solve for k. This should give us 2(3) + 5(-4) - 6(1) - 3(-2) = k, which simplifies to -2 + (-20) - 6 + 6 = k, or k = -22.

It seems like you may have made a small mistake in your calculations, as I am also getting a value of k = -22 instead of -26 or -14. I would recommend double checking your calculations and making sure you are using the correct signs for each term.

In summary, the equation of the hyperplane H in R4 that passes through P(3, -4, 1, -2) and is normal to u = [2, 5, -6, -3] is 2x1 + 5x2 - 6x3 - 3x4 = -22. I hope this helps clarify any confusion and good luck with your studies in linear algebra!

What is a Simple Hyperplane Equation?

A simple hyperplane equation is a mathematical representation of a hyperplane, which is a flat surface that divides a space into two parts. It is expressed as Ax + By + Cz + D = 0, where A, B, and C are the coefficients of the variables x, y, and z, and D is a constant.

What are the applications of Simple Hyperplane Equations?

Simple hyperplane equations are used in various fields, such as machine learning, computer graphics, and optimization. In machine learning, they are used to separate data points into different classes. In computer graphics, they are used to define the boundaries of 3D objects. In optimization, they are used to find the optimal solution to a problem.

What is the difference between a Simple Hyperplane Equation and a Linear Equation?

A simple hyperplane equation is a special case of a linear equation in three variables. It represents a flat plane in three-dimensional space, while a linear equation can represent a line in one, two, or three dimensions. Additionally, a simple hyperplane equation has four terms (Ax + By + Cz + D), while a linear equation has two terms (mx + b).

What am I missing if I only have two variables in my Simple Hyperplane Equation?

In a simple hyperplane equation, the number of variables (x, y, z) should match the dimension of the space (1, 2, or 3 dimensions). If there are only two variables, it means that the equation represents a line in two-dimensional space. To represent a hyperplane, there should be three variables.

Can a Simple Hyperplane Equation have more than three variables?

No, a simple hyperplane equation can only have three variables (x, y, z) because it represents a flat plane in three-dimensional space. If there are more than three variables, it means that the equation represents a hyperplane in a higher-dimensional space, which is not possible in a simple hyperplane equation.

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