find the distance between the point 𝐒 and the line 𝔏 with the given equation. 𝐒(4,-3) ; x-8 y+5=

Find the distance between the point 𝐒 and the line 𝔏 with...

Key Concepts

Distance from a Point to a Line

This concept refers to the shortest distance between a given point and a line, which is measured along the perpendicular from the point to the line. The standard formula for this distance is |Ax? + By? + C| / ?(A² + B²), where (x?, y?) are the coordinates of the point and Ax + By + C = 0 is the equation of the line. This formula encapsulates both the use of absolute value to ensure a non-negative distance and the normalization of the coefficient vector.

Standard Form of a Line Equation

This concept involves expressing the equation of a line in the form Ax + By + C = 0. This format is crucial in analytic geometry because it allows for a systematic approach to solving problems such as finding the distance between a point and the line, determining parallelism or perpendicularity between lines, and more.

Transcript

00:01 We're going to calculate the distance between the point p, which is 1 -9 -1, and the line 4x plus 5y equals 1.

00:26 I'm going to follow the example in the book, but also going to deviate a little from it.

00:37 First thing i'm going to do is i'm going to solve this line for y.

00:42 So 5y equals negative 4x plus 1, and y, dividing by 5 on both sides, is negative 5ths, oops, 4 fifths, plus 1 fifth, i divided by 5 on both sides.

01:11 So i can see that the slope of this line is negative four -fifths, and i'm going to call that b over a.

01:23 So b is negative four, and a is four -fifths.

01:28 Now, we can make a parametric set of parametric expressions for this line using x0 plus at and y0 plus at.

01:51 And y0 plus a t.

01:56 So we just need a point to call x0 y zero.

02:00 So i'm going to let x zero equal one.

02:11 I could have chose zero or any other number, but i'm deciding to choose one.

02:18 And so if x -0 equals 1, then y -0 will be negative 4 -fifths times 1 plus 1 -5th, which evaluates to negative 3 -5ths.

02:40 And so now i can write out the parametric expressions.

02:50 1 plus a, a is 1, t.

03:01 No, wait a minute, a is 5, a is a denominator, a is 5, 1 plus 5, t, comma, negative 3 5ths, minus 40, because b is negative 4.

03:30 All right.

03:33 So now if i set t equal to 1, number, i'm going to set t equal to zero.

03:45 Excuse me.

03:47 Then we get a point on the line, which i'm going to call q, which is 1 plus 5 times zero.

03:57 That's just 1.

04:06 And another point would be negative 3 fifths.

04:17 And this point lies on line l with a parallel.

04:38 Vevtv, which is 5, negative 4.

04:56 And we could actually get that from the slope.

05:04 I'm trying to draw here.

05:05 Negative 4 would go down here, and 5 would go down here.

05:14 So run, comma, rise.

05:17 And we actually could have gotten this point, just by setting x equals 1 and then y would be 3 5th so we could go directly there so in theory we don't really or even in reality we don't really have to create the parametric equation okay now from q to p we have the vector to p we have the vector which is just going to be p minus q so one minus one is zero negative one minus negative one minus three -fifths that's the same as negative one plus three -fifths is going to be negative two -fifths all right so now we have vector w v, q, and p.

07:00 And that's really all we need.

07:12 So the book lists it as p, vector p, vector vector v, vector v, is the inner product of w and v, over the norm of v squared.

07:47 Times vector v, which is the inner product of w and v over the inner product of v and v, because the norm of v squared is just the inner product of v and v times vector v.

08:08 Interproduct of w and v...

Please give Ace some feedback