Two parameterized lines are given. Are they the same line? r⃗1(t)=(5-3 t) i⃗+(1+t) j⃗+2 t k⃗

step 1 So we have two parameterized lines here. We want to see if they are secretly the same line. I'm

Two parameterized lines are given. Are they the same line? ...

Two parameterized lines are given. Are they the same line?
$$\begin{aligned}
&\vec{r}_{1}(t)=(5-3 t) \vec{i}+(1+t) \vec{j}+2 t \vec{k}\\
&\vec{r}_{2}(t)=(2+6 t) \vec{i}+(2-2 t) \vec{j}+(2-4 t) \vec{k}
\end{aligned}$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Parametric Representation of a Line

The parametric representation expresses a line by specifying a fixed point on the line and a direction vector, using a parameter that scales this direction vector. This form is critical in analyzing and comparing lines because it directly shows how points on the line are generated and allows for operations such as checking intersections or coincidences.

Direction Vectors

The direction vector indicates the orientation of the line. It is obtained from the coefficients of the parameter in the parametric equation. By comparing the direction vectors (up to a scalar multiple), one can determine if the lines are parallel, which is a necessary condition for them to be the same line.

Point Membership Test

Checking whether a specific point on one line lies on the other involves substituting the coordinates of that point into the equation of the other line and solving for the parameter. This test verifies if there is an overlap between the lines beyond just having parallel directions.

Line Coincidence Criteria

For two lines to be identical, they must not only be parallel (their direction vectors are scalar multiples) but also share at least one common point. This combination ensures that the entire set of points constituting one line is exactly contained within the other.

Transcript

00:01 So we have two parameterized lines here.

00:02 We want to see if they are secretly the same line.

00:05 I'm going to rewrite these as a sum of two components, the t component...

00:11 Component is the wrong word.

00:13 The t part and the not t part.

00:16 It could be 5i plus j plus negative 3i plus j plus 2k times t.

00:29 So i've taken out all the constant parts of each of these components and put them over here and then separated from that all of the stuff being multiplied by t.

00:40 We do the same thing for r2 here.

00:43 2i plus 2j plus 2k plus 6i minus 2j minus 4k times t.

00:59 What we see here is that these two lines are parallel because one of these directional components, that is the part that's being multiplied by t, is just a scalar multiple of the other.

01:14 You can multiply the top guy by negative 2 and you get the bottom guy.

01:20 It comes down whether to...

01:22 Because they're parallel, they're equal if and only if they have a point in common.

01:27 So let's see.

01:28 We know that if we set t equal to 0, this guy goes away and we have that r1 of 0 equals 5i plus j.

01:40 Can we get 5i plus j in r2? well, r2 of t, if we want this to be 5i plus j, then in particular we see that there must be 0 on our k component...

Please give Ace some feedback