r1(t) = t, 2 − t, 15 + t2 and r2(s) = 5 − s, s − 3, s2At what point do the curves intersect?(x, y, z) = Find their angle of intersection, ?, correct to the nearest degree.? = At what point do the curves r₁(t) = (t, 2 − t, 15 + t²) and r₂(s) = (5 – s, s − 3, s²) intersect?(x, y, z) =Find their angle of intersection, 0, correct to the nearest degree.0 =O

Answered: At what point do the curves r₁(t) = (t,…

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

r₁(t) = t, 2 − t, 15 + t²and r₂(s) = 5 − s, s − 3, s²

At what point do the curves intersect?

(x, y, z) =

Find their angle of intersection, ?, correct to the nearest degree.

? =

At what point do the curves r₁(t) = (t, 2 − t, 15 + t²) and r₂(s) = (5 – s, s − 3, s²) intersect?
(x, y, z) =
Find their angle of intersection, 0, correct to the nearest degree.
0 =
O

Expert Solution

Step 1

Given curves:

$r_{1} (t) = <t, 2 - t, 15 + t^{2}>$

$r_{2} (s) = <5 - s, s - 3, s^{2}>$

To find:

a) The point of intersection of both the curves.

b) Angle between them.

Angle between the direction vectors $\vec{u} a n d \vec{v}$ :

$θ = \cos^{- 1} (\frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|})$

Here, $\vec{u} \cdot \vec{v}$ denotes the dot product of the two vectors.

It is defined as $\vec{u} \cdot \vec{v} = a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}$ , where $\vec{u} = <a_{1}, b_{1}, c_{1}>$ and $\vec{v} = <a_{2}, b_{2}, c_{2}>$ .

$|\vec{u}| = \sqrt{{(a_{1})}^{2} + {(b_{1})}^{2} + {(c_{1})}^{2}}$ and $|\vec{v}| = \sqrt{{(a_{2})}^{2} + {(b_{2})}^{2} + {(c_{2})}^{2}}$ .

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At what point do the curves r₁(t) = (t, 2 − t, 15 + t²) and r₂(s) = (5 – s, s − 3, s²) intersect? (x, y, z) = Find their angle of intersection, 0, correct to the nearest degree. 0 = O