give answers to the following twenty mathematics questions you may han

Question

Give answers to the following twenty mathematics questions. You may handwrite or typeset your answers but you must submit your answers as a PDF le via the ECS submission system.You are given the following vectors and matrices: \mathbf{a}=\left[\begin{array}{l}
1 \\
4 \\
8
\end{array}\right] \quad \mathbf{b}=\left[\begin{array}{c}
8 \\
-4 \\
1
\end{array}\right] \mathbf{c}=\left[\begin{array}{c}
2 \\
-2 \\
1
\end{array}\right] \quad \mathbf{d}=\left[\begin{array}{l}
8 \\
0 \\
6
\end{array}\right] \mathbf{A}=\left[\begin{array}{ccc}
0 & 5 & 0 \\
-5 & 3 & 0 \\
-1 & 0 & 2
\end{array}\right] \quad \mathbf{B}=\left[\begin{array}{lll}
3 & 5 & 0 \\
0 & 4 & 0 \\
0 & 0 & 1
\end{array}\right] \quad \mathbf{C}=\left[\begin{array}{lll}
1 & 0 & 2 \\
0 & 1 & 2 \\
0 & 0 & 1
\end{array}\right] 1. а +b 2. с +d 3. За 4. -2b 5. а — b 6. a 7. |Ы 8. а-b 9. с -d 10. What is the angle betwcen vectors a and b? 11. What is the angle between vectors c and d? 12. How long is the projection of vector c onto vector d? \text { 13. Calculate } \mathbf{e} \text {, the linear interpolation betwecn } \mathbf{c} \text { and } \mathbf{d}, \mathbf{e}=(1-t) \mathbf{c}+t \mathbf{d} \text {, for } t=0.8 \text {. } 14. Ab 15. Вс 16. А + В 17. АВ 18. ВС 19. What two-dimensional transformation is represented by the 3 x 3 matrix C? 20. Give a 3 × 3 matrix that represents a rotation in two-dimensional space of 60°.