Question

Hello,

Can somebody help me with the following two problems?

1. Find a value of a such that the vector <a, 1,1> makes an angle of 45 degrees with the vector <1,2,1> or show that no such a exists.

---I started by using theta = arccos (u*v)/(absu*absv) must equal (sqrt2)/2 and attempting to solve from there...but I'm not sure if that's the best way to proceed.

2. Let u = 3i + j; v = 5i - 2j; and w = i - j. Find scalars, a and b such that u = av +bw.

I have no idea how to proceed with this one.

I would greatly appreciate any help!!!

Answer 1

u = <a, 1,1>

v = <1,2,1>

u dot v = 1/2 sqrt(2) |u| |v| ---->

(u dot v)^2 = 1/2 |u|^2 |v|^2 ---->

(a+3)^2 = 1/2 (a^2 + 2)*6 ---->

2 a^2 -6 a -3 = 0


Let u = 3i + j; v = 5i - 2j; and w = i - j. Find scalars, a and b such that u = av +bw.

av+bw =

(5a +b)i + (-2a-b)j

av+bw = u --->

5a + b = 3

2a + b = -1