If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Precalculus (Eureka Math/EngageNY)>Unit 2

Lesson 7: Topic E: First-person video games—projection matrices

# 4. Using the line equation

Vertical lines have an undefined slope, so how can we solve for the intersection point? Instead of using slope-intercept form, we need to use the line equation, or implicit form. Click here to see why dividing by zero is bad. Click here to try the interactive program shown in the video.

## Want to join the conversation?

• I have heard the term implicit before; what does it mean?
(1 vote)
• An explicit equation shows a clear relation between x and y, such as y=3x+7. This way it is very clear what the relationship between x and y is. If x=2, then it is easy to compute y=3(2)+7=13. An implicit equation, on the other hand, represents an implied relationship between x and y, such as 5x+2y=7. It is not as obvious that y=-4 when x=3.

In everyday use, I might say I prefer not working with numbers, which might be an implicit way of saying I don't like math.
• How do you do this (ray/line intercept) with a line drawn in 3D space? It seems like the slope intercept form would not be usefull in this case?
• In 3D the ray intersects with 2D objects, not lines.
(1 vote)
• At A(2.973,2) B(2.973,-1) => AB: 3x-8.92=0 (vertical line case)
I just can't figure it out, how c has become -8.92 from the Δyx-Δxy+Δxi=0 <=> ax+by+c=0 identity. I know it is true, but this conversion not takes me there.
If c=Δx*i, the constant would be 0 * i = 0! There is no change in x, this way only 3x=0 remains. What did I miss out?
(1 vote)
• It's a good question, which was glossed over. The issue is that you need to find out what `iΔx` is, but `Δx` is 0 and `i` is undefined since there is no intercept for a vertical line. And yet we can still do it.

Remember one way to write the equation of a line is:
`y = (Δy/Δx)x + i`

If we multiply by `Δx` (so we can avoid dividing by 0), then we get
`yΔx = xΔy + iΔx`

So `iΔx = yΔx - xΔy`.

For the case A = (2.973, 2) and B = (2.973, -1), `Δx = 0` and `Δy = 3`, so we have:
`iΔx = 0y - 3x = -3x` - notice we are keeping the left hand side as `iΔx` since this is what we want to calculate.
Then we just plug in a value for `x` that we know, maybe `Ax`, but it doesn't matter since all `x` on the line will be the same.

`Ax = 2.972`, so `iΔx = -3(2.972) = -8.916` (-8.92 when rounded).
So now `xΔy - yΔx + iΔx = 0` becomes `3x - 0y + (-8.92) = 0`.
Hence `3x - 8.92 = 0`

This is why we multiply by `Δx` so we can get numbers we can deal with rather 1/0 multiplied by 0.
• how do you find the y-intercept given 2 points without having to divide by zero if there is no change in x?
(1 vote)
• If there is no change in x then the line will be vertical, and so parallel to the y-axis. Parallel lines don't intersect so it's not possible to find the y-intercept for a vertical line.
(1 vote)
• i thot i was just going to animate
(1 vote)
• At time ,how did the y intercept became 11 ? please solve this !
(1 vote)
• i have a question y do we have to do this