# How many planes can be made to pass through three distict points if they are – a-collinear b-non-collinear what is the answer?

If the points are colinear then an infinite number of planes can be made to pass through them. If three distinct points are non-colinear then exactly one plane passes through them.

Let us look at the general case.

Haven’t Found The Relevant Content? Hire a Subject Expert to Help You With
How many planes can be made to pass through three distict points if they are – a-collinear b-non-collinear what is the answer?

The general equation of a plane is ##ax+by+cz = d## for constants ##a##, ##b##, ##c##, ##d##.

For a given plane, this equation is unique only up to a constant factor. That is, the same plane is also described by the equation:

##kax+kby+kcz=kd## for any ##k != 0##

Let the three points be ##(x_1, y_1, z_1)##, ##(x_2, y_2, z_2)## and ##(x_3, y_3, z_3)##.

Then we have a system of three linear equations:

##ax_1 + by_1 + cz_1 = d##

##ax_2 + by_2 + cz_2 = d##

##ax_3 + by_3 + cz_3 = d##

In the general case, this is a little painful to deal with using substitution and/or elimination, so permit me to show how to do it using matrix arithmetic.

Let ##M = ((x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3))##

Then our three equations become:

##M((a), (b), (c)) = ((d), (d), (d))##

If we can find ##M^(-1)## then we get:

##((a), (b), (c)) = M^(-1)M((a), (b), (c)) = M^(-1)((d), (d), (d))##

Let:

##X_1 = det((y_2, z_2), (y_3, z_3)) = y_2z_3 – z_2y_3####Y_1 = z_2x_3 – z_3x_2####Z_1 = x_2y_3 – y_2x_3####X_2 = y_3z_1 – z_3y_1####Y_2 = z_3x_1 – x_3z_1####Z_2 = x_3y_1 – y_3x_1####X_3 = y_1z_2 – z_1y_2####Y_3 = z_1x_2 – x_1z_2####Z_3 = x_1y_2 – y_1x_2##

The determinant of ##M## is given by the formula:

##det(M) = x_1X_1 + y_1Y_1 + z_1Z_1##

If ##det(M) != 0## then ##M^-1## is given by:

##M^(-1) = 1/det(M)((X_1, X_2, X_3), (Y_1, Y_2, Y_3), (Z_1, Z_2, Z_3))##

How can this go wrong?

If the points are not distinct, then two of the rows of ##M## will be identical and ##det(M) = 0##

If the points are colinear, then ##det(M) = 0##

If the plane also passes through ##(0, 0, 0)## then ##det(M) = 0##

This last case is the one where ##d = 0##, so ##M^(-1)((d),(d),(d))## would be fairly useless anyway.

## Calculate the price of your order

Select your paper details and see how much our professional writing services will cost.

We`ll send you the first draft for approval by at
Price: \$36
• Freebies
• Format
• Formatting (MLA, APA, Chicago, custom, etc.)
• Title page & bibliography
• Amendments to your paper when they are needed
• 275 word/double-spaced page
• 12 point Arial/Times New Roman
• Double, single, and custom spacing

Our custom human-written papers from top essay writers are always free from plagiarism.

Your data and payment info stay secured every time you get our help from an essay writer.

Your money is safe with us. If your plans change, you can get it sent back to your card.

## How it works

1. 1
You give us the details
Complete a brief order form to tell us what kind of paper you need.
2. 2
We find you a top writer
One of the best experts in your discipline starts working on your essay.
3. 3
You get the paper done

## Samples from our advanced writers

Check out some essay pieces from our best essay writers before your place an order. They will help you better understand what our service can do for you.

## Perks of our essay writing service

We offer more than just hand-crafted papers customized for you. Here are more of our greatest perks.

• Swift delivery
Our writing service can deliver your short and urgent papers in just 4 hours!
• Professional touch
We find you a pro writer who knows all the ins and outs of your subject.
• Easy order placing/tracking
Create a new order and check on its progress at any time in your dashboard.
• Help with any kind of paper
Need a PhD thesis, research project, or a two-page essay? For you, we can do it all.
• Experts in 80+ subjects
• Calculations and code
We also do math, write code, and solve problems in 30+ STEM disciplines.

• ### What if I’m dissatisfied with the paper I get?

The average quality score at our professional custom essay writing service is 8.5 out of 10. The high satisfaction rate is set by our Quality Control Department, which checks all papers before submission. The final check includes:
• Compliance with initial order details.
• Plagiarism.
• Proper referencing.
If for some reason we happen to leave a mistake unnoticed, you are invited to request unlimited revisions of your custom-written paper. For more information, check our Revision Policy. We will do our best to make your experience with Familiar Essays enjoyable.
• ### I need an essay on the same day. Is it something you can do?

Sure. Our writing company offers a fast service with an 8-hour deadline for orders up to master’s level. Make sure to specify the deadline in the order form and our writers will write a paper within the indicated timeslot. Just proceed to submit your requirements here Once you order a custom-written essay, our managers will assign your order to the well-suited writer, who has the best skills and experience for preparing your specific assignment. You can also request one of these extra features:
• Choose the Writer’s Samples option – study 3 randomly-provided pages from orders that have been written by the assigned writer.
• Request a specific writer – choose an academic writer from the dropdown list in the order’s form (optional for returning customers).
You can be sure that your custom writing order will be accomplished by one of our 400+ professional academic writers. They all pass a series of tests to prove their writing prowess and hold the reputation of being the most professional in the industry. Want to make sure writer’s skills match your needs? Get more details on how to choose the appropriate author.
• ### How can I be sure your writing service is not a scam?

We understand that a shade of mistrust has covered the paper writing industry, and we want to convince you of our loyalty. Apart from high-quality writing services, we offer:
• The chances of students to boost writing skills in a quick and effective way.
• The opportunity to manage studies and free time in an enjoyable manner.
• The possibilities to improve overall academic performance.
Our custom writing company has been working for more than 12 years and always puts quality and clients’ needs first. Our operations are legally documented, we are easily accessible online and offline,
• ### Is it legal to use your professional writing service?

Yes. Custom writing help is not prohibited by any university or college. It’s a 100% legal way of getting professional assistance with paper writing. Hiring writers from an essay writing company is in many ways similar to consulting a tutor – we help you solve the writing issues at hand.