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!!!

2. Originally Posted by riverjib
Hello,

Can somebody help me with the following two problems?
...

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!!!
$\displaystyle u = 3i + j = a(5i - 2j) + b(i - j) = av + bw$

$\displaystyle \Longleftrightarrow (1) \,5ai + bi = 3i$ and $\displaystyle (2) \, -2aj - bj = j$

from (2) $\displaystyle b = -2a$
substitute to (1), $\displaystyle 5a + -2a = 3$ or $\displaystyle a = 1$

so, $\displaystyle b = -2$

3. For #1, solve for a in:
$\displaystyle \frac{{\left\langle {a,1,1} \right\rangle \cdot \left\langle {1,2,1} \right\rangle }}{{\sqrt 6 \sqrt {a^2 + 2} }} = \frac{{a + 3}}{{\sqrt 6 \sqrt {a^2 + 2} }} = \frac{{\sqrt 2 }}{2}$