Linear Math: Find a and b for Point on Line

In summary, linear math, also known as linear algebra, involves finding solutions to systems of linear equations and understanding the relationships between variables. To find the values of a and b for a point on a line, the equation of a line is used, which is y = ax + b. This allows us to determine the slope and intercept of the line and is important for many applications in math and science. While there are other methods for finding a and b, the most commonly used method is using the equation of a line.
  • #1
~Sam~
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Homework Statement




Find a and b such that the point (a,b,0) lies on the line passing through (-1,-1,6) and (-9,7,2)

Homework Equations



Basic line equations, from P to Q= Q-P..etc

The Attempt at a Solution



I keep getting the wrong answer, but I got a point at exactly P (-1,-1,6) and a direciton by doing Q-P...and solving for z=0...
 
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  • #2
so the line equation is
L(t) = P + t*(Q-P)

show your work, but soudns like you're heading in the right direction, solve for t, when z=0 as you say, then substitute back into line
 
  • #3


The first step in solving this problem is to find the direction vector of the line passing through the two given points. This can be done by subtracting the coordinates of one point from the other. In this case, we have (-9,7,2) - (-1,-1,6) = (-8,8,-4). This vector represents the direction of the line.

Next, we can use the general equation of a line in 3D, which is (x,y,z) = (x0,y0,z0) + t(a,b,c), where (x0,y0,z0) is a point on the line and (a,b,c) is the direction vector. In this case, we have (x,y,z) = (-1,-1,6) + t(-8,8,-4).

To find the values of a and b for the given point (a,b,0), we can equate the first two coordinates of the equation (x,y,z) = (-1,-1,6) + t(-8,8,-4) to the coordinates of the given point. This gives us the equations:

a = -1 - 8t
b = -1 + 8t

Solving these equations simultaneously, we get t = -a/8 and t = (b+1)/8. Setting these two expressions equal to each other and solving for a, we get a = -b-2.

Therefore, the values of a and b for the point (a,b,0) to lie on the line passing through (-1,-1,6) and (-9,7,2) are a = -b-2 and b = -1 + 8t.
 

1. What is linear math?

Linear math, also known as linear algebra, is a branch of mathematics that deals with the study of linear equations and their properties. It involves finding solutions to systems of linear equations and understanding the relationships between variables.

2. How do you find the value of a and b for a point on a line?

To find the values of a and b for a point on a line, you will need to use the equation of a line, which is y = ax + b. Plug in the coordinates of the given point for x and y, and then solve for a and b using algebraic methods.

3. What is the importance of finding a and b for a point on a line?

Finding the values of a and b for a point on a line allows us to understand the relationship between the given point and the line. It also helps us to determine the slope and intercept of the line, which are important for many applications in math and science.

4. Can you find a and b for any point on a line?

Yes, you can find the values of a and b for any point on a line as long as the point is not on the vertical line x = 0. This is because the equation of a line only works for non-vertical lines, as the slope would be undefined for a vertical line.

5. Are there any other methods for finding a and b for a point on a line?

Yes, there are other methods for finding a and b for a point on a line, such as using the point-slope form of a line or using matrices and Gaussian elimination. However, the most commonly used method is using the equation of a line, y = ax + b.

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