1. ## Linear Algebra Help

These are linear algebra problems for an advanced engineering math class.
1. Find the angles in the triangle with these vertices:
[2, -1, 0] , [5, -4, 3] , and [1, -3, 2]

2. Determine the value a so that vectors
x = 2i + aj + k and y = 4i - 2j - 2k are perpendicular. Compute x(dotproduct)y to verify the result.

3. Compute the inner product of the following functions on the interval [-pi, pi] with n and m distinct positive integers:
a. <sin mx, sin nx>
b. <cos mx, cos nx>
c. <cos mx, sin nx>
d. <cos nx, cos nx>
e. <sin nx, cos nx>
Which functions are orthogonal?

Any help would be greatly appreciated. Thanks in advance.

2. 1. This is simple, let's say we label the vectors:

$\displaystyle \vec{x} = <2, -1, 0>$

$\displaystyle \vec{y} = <5, -4, 3>$

$\displaystyle \vec{z} = <1, -3, 2>$

Now, the formula for the $\displaystyle cos\theta$ is:

$\displaystyle cos\theta_1 = \frac{\vec{x}\cdotp\vec{y}}{\lVert\vec{x}\rVert*\l Vert\vec{y}\rVert}$

$\displaystyle cos\theta_2 = \frac{\vec{y}\cdotp\vec{z}}{\lVert\vec{y}\rVert*\l Vert\vec{z}\rVert}$

$\displaystyle cos\theta_3 = \frac{\vec{x}\cdotp\vec{z}}{\lVert\vec{x}\rVert*\l Vert\vec{z}\rVert}$

All you have to do is plug and chug:

$\displaystyle \color{red}\vec{x}\cdotp\vec{y}$$\displaystyle = 2*5 + -1*-4 + 0*3 = 10 + 4 = \color{blue}14 \displaystyle \color{red}\vec{y}\cdotp\vec{z}$$\displaystyle = 5*1 + -4*-3 + 3*2 = 5 + 12 + 6 = \color{blue}23$

$\displaystyle \color{red}\vec{x}\cdotp\vec{z}$$\displaystyle = 2*1 + -1*-3 + 0*2 = 2 + 3 = \color{blue}5 \displaystyle \color{red}\lVert\vec{x}\rVert$$\displaystyle = \sqrt{2^2 + (-1)^2 + 0^2} = \color{blue}\sqrt{5}$

$\displaystyle \color{red}\lVert\vec{y}\rVert$$\displaystyle = \sqrt{5^2 + (-4)^2 + 3^2} = \color{blue}\sqrt{50} \displaystyle \color{red}\lVert\vec{z}\rVert$$\displaystyle = \sqrt{1^2 + (-3)^2 + 2^2} = \color{blue}\sqrt{14}$

You can solve from there I'm sure.

3. 2. Determine the value a so that vectors
x = 2i + aj + k and y = 4i - 2j - 2k are perpendicular. Compute x(dotproduct)y to verify the result.
2 vectors are perpendicular if their dot product is 0.

3. Compute the inner product of the following functions on the interval [-pi, pi] with n and m distinct positive integers:
a. <sin mx, sin nx>
b. <cos mx, cos nx>
c. <cos mx, sin nx>
d. <cos nx, cos nx>
e. <sin nx, cos nx>
Which functions are orthogonal?
on the interval [a,b], the inner product of f and g is $\displaystyle \int_a^b f(x)g(x)dx$. Functions are orthogonal if their inner product is 0.