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°.  Fig: 1  Fig: 2  Fig: 3  Fig: 4  Fig: 5  Fig: 6  Fig: 7  Fig: 8  Fig: 9  Fig: 10  Fig: 11  Fig: 12  Fig: 13  Fig: 14  Fig: 15  Fig: 16  Fig: 17  Fig: 18  Fig: 19  Fig: 20  Fig: 21  Fig: 22  Fig: 23