Vector calculations, operators, matrices

Question

1) A plain goes through the origo (0, 0, 0) and it is perpendicular to the vector seen on the picture.enter image description here What is the P matrix, that is projected to the plain, and what is the mirroring T matrix? Their size is 3x3.

2) tan(φ)=1/50. Calculate the 2x2 size matrix of the operator that rotates the plain by φ angle.

3) Let φ angle be the same as in task 2). Calculate the 3x3 size matrix of the linear operator that rotates v vector (seen on the picture ) by φ angle.

If the picture is not seen v is a column vector having (7, 6, 5) with 7 on the top.

I would like to understand how to solve these exercises, so if someone knows the answer, please be kind to explain it to me, because it would be a life saver. Thank you!!!

Answer 1

(1) There is a formula on wikipedia. It says if you can find an orthonormal basis $u_1,u_2$ of the plane, and you make the matrix $A$ whose columns are $u_1$ and $u_2$, then the projection matrix is $P = AA^T$. So if we can find some $u_1$ and $u_2$, you can do the rest using the formula. First, find any old vector in the plane. So just find something whose dot product with $(7,6,5)$ is zero. For example, we can use $(1,-2,1)$. If we take the cross product of this vector with $(7,6,5)$, we will get another vector in the same plane, which is orthogonal to $(1,-2,1)$. So compute $(1,-2,1) \times (7,6,5) = 2(-8,1,10)$. Now just re-scale these vectors so they have length one, and we can use these as $u_1$ and $u_2$: $$\begin {align*} u_1 &= \frac{1}{\sqrt{6}} (1,-2,1) \\ u_2 &= \frac{1}{\sqrt{165}} (-8,1,10) \end {align*}$$ Your matrix $A$ is then $$ \begin{pmatrix} \frac{1}{\sqrt{6}} & \frac{-8}{\sqrt{165}} \\ \frac{-2}{\sqrt{6}} & \frac{1}{\sqrt{165}} \\ \frac{1}{\sqrt{6}} & \frac{10}{\sqrt{165}} \end{pmatrix} $$ Finally just use the formula $P = AA^T$.

(2) A rotation matrix that rotates counter-clockwise by angle $\phi$ is: $$ \begin{pmatrix} \cos(\phi) & -\sin(\phi) \\ \sin(\phi) & \cos(\phi) \end{pmatrix} $$ See for instance: wikipedia

(3) You need to specify an axis of rotation for $3 \times 3$. I'm not sure what you mean? Do you mean that $v = (7,6,5)$ is the axis of rotation? Or that the axis is orthogonal to this vector? In any case, if you know the correct axis, see the wikipedia page for the formula for the $3 \times 3$ rotation matrix.

Answer 2

I solved the following system of equations 3 times with different parameters: In the following I calculate the image of the first unit vector (denoted by x):

$x_{1}+h_{1}=1;$

$ x_{2}+h_{2}=0; $

$ x_{3}+h_{3}=0; $

$ x_{1}*h_{1}+x_{2}*h_{2}+x_{3}*h_{3}=0; $ (because x and h are perpendicular to each other)

$ -7/5*x_{1}-6/5*x_{2}=x_{3}; $ (because x must be on the plain)

$ (x_{1}^{2}+x_{2}^{2}+x_{3}^{2})+(h_{1}^{2}+h_{2}^{2}+h_{3}^{2})=1 $ (Using the Pythagorean theorem the (length of the x vector)^2+(length of the m vector)^2 must be 1^2)

Using the same method I want to find the image of the 2nd unit vector.

$ x_{1}+h_{1}=0;$

$ x_{2}+h_{2}=1;$

$x_{3}+h_{3}=0;$

$ x_{1}*h_{1}+x_{2}*h_{2}+x_{3}*h_{3}=0; $

$ -7/5*x_{1}-6/5*x_{2}=x_{3}; $

$ (x_{1}^{2}+x_{2}^{2}+x_{3}^{2})+(h_{1}^{2}+h_{2}^{2}+h_{3}^{2})=1 $

And for the image of the 3rd unit vector:

$ x_{1}+h_{1}=0; $

$ x_{2}+h_{2}=0; $

$ x_{3}+h_{3}=1; $

$ x_{1}*h_{1}+x_{2}*h_{2}+x_{3}*h_{3}=0; $

$ -7/5*x_{1}-6/5*x_{2}=x_{3}; $

$ (x_{1}^{2}+x_{2}^{2}+x_{3}^{2})+(h_{1}^{2}+h_{2}^{2}+h_{3}^{2})=1 $

Using wolfram alpha I solve these systems, and for the first unit vector its working, but for the others its not! What is the problem? Is it a good way to solve this problem anyway?