I am struggling with a couple of problems and I hope someone can help me. Here are the questions:
- a) In an orthonormal basis $e_1, e_2$ in the plane, we choose a new basis $\hat{e}_1, \hat{e}_2$ where $$ \left\{\begin{array}{l} \hat{ e }_1=2 e _1+3 e _2 \\ \hat{ e }_2=- e _1+ e _2 \end{array}\right. $$ I need to calculate the scalar product $\hat{ u } \bullet \hat{ v }$ of the vectors $\hat{ u }=(1,1)$ and $\hat{ v }=(-2,1)$ in the new basis $\hat{e}_1, \hat{e}_2$.
b) A tetrahedron has vertices at points $P_0:(0,0,0), P_1:(1,1,1), P_2:(0,1,2)$ and $\left.P_3\right):(2,-2,0)$. The orthogonal projection of the tetrahedron onto the plane $-x+2 y-z=0$ forms a parallelogram. I need to find the area of this parallelogram.
For part a), I tried to use the fact that the scalar product is preserved under a change of basis. So I calculated the coordinates of $\hat{ u }$ and $\hat{ v }$ in the original basis $e_1, e_2$, and then used the formula for the scalar product in that basis. However, I did not get the correct answer, so I must have made a mistake somewhere.
For part b), I am not sure where to start. I know that the projection of a point onto a plane is obtained by subtracting the orthogonal projection onto the normal vector of the plane, but I am not sure how to apply this to a whole tetrahedron.
Any help or guidance would be greatly appreciated!