(Solved):   2.1. Write a unit vector in matrix form that describes the direction of the cross product ...

 

2.1. Write a unit vector in matrix form that describes the direction of the cross product of \( \mathbf{p}=5 \mathbf{i}+3 \mathbf{k} \) and \( \mathbf{q}=3 \mathbf{i}+4 \mathbf{j}+5 \mathbf{k} \). Problem 2.3 Vectors \( \mathbf{p}=2 \mathbf{i}+3 \mathbf{j}+5 \mathbf{k} \) and \( \mathbf{q}=3 \mathbf{i}+6 \mathbf{k} \) are given. Find a vector \( \mathbf{r} \) that is perpendicular to both. 2.5. The following frame \( B \) was moved a distance of \( d=(5,2,6)^{T} \). Find the new location of the frame relative to the reference frame. \[ B=\left[\begin{array}{cccc} 0 & 1 & 0 & 2 \\ 1 & 0 & 0 & 4 \\ 0 & 0 & -1 & 6 \\ 0 & 0 & 0 & 1 \end{array}\right] \] Problem 2.6 The following frame \( B \) was moved a distance of \( d=[4,2,6]^{T} \). Find the new location of the frame relative to the reference frame. \[ B=\left[\begin{array}{cccc} 0 & 1 & 0 & 2 \\ 1 & 0 & 0 & 5 \\ 0 & 0 & -1 & 8 \\ 0 & 0 & 0 & 1 \end{array}\right] \] Problem \( 2.7 \) For frame \( F \), find the values of the missing elements and complete the matrix representation of the frame. \[ F=\left[\begin{array}{cccc} ? & 0 & -1 & 4 \\ ? & 0 & 0 & 5 \\ ? & -1 & 0 & 7 \\ 0 & 0 & 0 & 1 \end{array}\right] \] Problem \( 2.8 \) Find the values of the missing elements of frame \( B \) and complete the matrix representation of the frame. \[ B=\left[\begin{array}{cccc} 0.707 & ? & 0 & 2 \\ ? & 0 & 1 & 4 \\ ? & -0.707 & 0 & 5 \\ 0 & 0 & 0 & 1 \end{array}\right] \]