Simple Hyperplane Equation - What am I Missing?

[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 2_x1+5_x2-6_x3-3_x4 = 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?

 

[micromass]

It's probably a typo.

 

[Mythic]

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?

 

[cowmoo32]

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

 

[blue_raver22]

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!