I need help for problem 9. Thanks! masses at A, B, and C, the center of mass of that system is given by the intersections of the mediansof the triangle.9. (a) Let u, v e R2. Describe the vectors x su + tv, where s +t = 1. Pay particular attentionto the location of x when s > 0 and whent > 0.(b) Let u, v, w e Ri. Describe the vectors x ru +sV + tw, wherer+s +t = 1. Pay particularattention to the location of x when each of r, s, and t is positive.10. Suppose x, y e R" are nonparallel vectors. (Recall the definition on p. 3.)

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Description: I need help for problem 9. Thanks! masses at A, B, and C, the center of mass of that system is given by the intersections of the mediansof the triangle.9. (a) Let u, v e R2. Describe the vectors x su + tv, where s +t = 1. Pay particular attentionto t

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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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I need help for problem 9. Thanks!

masses at A, B, and C, the center of mass of that system is given by the intersections of the medians
of the triangle.
9. (a) Let u, v e R2. Describe the vectors x su + tv, where s +t = 1. Pay particular attention
to the location of x when s > 0 and whent > 0.
(b) Let u, v, w e Ri. Describe the vectors x ru +sV + tw, wherer+s +t = 1. Pay particular
attention to the location of x when each of r, s, and t is positive.
10. Suppose x, y e R" are nonparallel vectors. (Recall the definition on p. 3.)

Expert Solution

Step 1

For (a)

We are given that u, v e R2. And x = su + tv, where s t
t 1- s
х%3D su + (1 —s)v
х %3Ds (и — v) + v
1
So,
which implies that x is a vector which is a linear
v passing through v.
(which can be observed by putting the value of s 0)
combination of the vector u

Step 2

Now we will check the behaviour of x for values of s and t.

Step 3

If both s ≥ 0, t ≥ 0 and satisfying s + t = 1, then the position of x will lie on the u-v vector as shown in fig.

Step 4

Now for (b),

We are given that u, v e R3. And x
where rs+t = 1.
ru + sv + tw,
t 1-r-s
So,
x rusv + (1 -r-s)w
x r(u w) s(v -w)w
Which implies that x is a vector which is a linear combination of the
w passing through w. (which can be observed
0 and r 0)
vector u w and v
by putting the value of s

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