In each case, determine whether or not the lines have a single point of intersection. If they do, give an equation of a planecontaining them.a) r₁=(5t, 2t - 1, 3t 3) and r₂ = (t -6, -t + 5,t - 9)b) r₁ =(4t, −4t + 1, t − 5) and r₂ = (2t — 3, −t, −t − 1)(Express numbers in exact form. Use symbolic notation and fractions where needed. Enter DNE if lines do not intersect.)(a) the equation of the plane:(b) the equation of the plane:

a Given that r→1=5t,2t−1,3t−3 and r2→=t−6,−t+5,t−9. We have to find the equation of the plane. Now,…

Answered: In each case, determine whether or not…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

In each case, determine whether or not the lines have a single point of intersection. If they do, give an equation of a plane
containing them.
a) r₁
=
(5t, 2t - 1, 3t 3) and r₂ = (t -6, -t + 5,t - 9)
b) r₁ =
(4t, −4t + 1, t − 5) and r₂ = (2t — 3, −t, −t − 1)
(Express numbers in exact form. Use symbolic notation and fractions where needed. Enter DNE if lines do not intersect.)
(a) the equation of the plane:
(b) the equation of the plane:

Expert Solution

Step 1

$(a)$ Given that ${\vec{r}}_{1} = 〈5 t, 2 t - 1, 3 t - 3〉$ and $\vec{r_{2}} = 〈t - 6, - t + 5, t - 9〉$ .

We have to find the equation of the plane.

Now, ${\vec{r}}_{1} at t_{1} = 〈5 t_{1}, 2 t_{1} - 1, 3 t_{1} - 3〉$

And $\vec{r_{2}} at t_{2} = 〈t_{2} - 6, - t_{2} + 5, t_{2} - 9〉$

For intersection point

$\begin{matrix} 5 t_{1} = t_{2} - 6........ (1) \\ 2 t_{1} - 1 = - t_{2} + 5...... (2) \\ 3 t_{1} - 3 = t_{2} - 9........ (3) \end{matrix}$

Now, solve equation $(1), (2) a n d (3)$ , we get

$t_{1} = 0 and t_{2} = 6$

If we substitute $t_{1} = 0 and t_{2} = 6$ in equation $(3)$ , it satisfy the third equation.

Hence, ${\vec{r}}_{1}$ and $\vec{r_{2}}$ intersects each other.

Now at $t_{1} = 0$ , ${\vec{r}}_{1} = 〈0, - 1, - 3〉$ .

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In each case, determine whether or not the lines have a single point of intersection. If they do, give an equation of a plane containing them. a) r₁ = (5t, 2t 1,3t 3) and r₂ = (t−6, -t + 5, t - 9)