Answer to Question #148578 in Analytic Geometry for Jyotiramay Rout

Question

Denote by a, b and c the column vectors a = (1 2 3), b = (-2 1 -3), c = (-2 -1 1) Calculate 2a - 5b, 2a- 5b +c, a'.b,

Answer 1

$$a=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, b=\begin{pmatrix} -2 \\ 1 \\ -3 \end{pmatrix}, c=\begin{pmatrix} -2 \\ -1 \\ 1 \end{pmatrix}$$

$$2a-5b=2\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}-5\begin{pmatrix} -2 \\ 1 \\ -3 \end{pmatrix}=\begin{pmatrix} 2(1)-5(-2) \\ 2(2)-5(1) \\ 2(3)-5(-3) \end{pmatrix}$$

$$=\begin{pmatrix} 12 \\ -1 \\ 21 \end{pmatrix}$$

$$2a-5b+c=2\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}-5\begin{pmatrix} -2 \\ 1 \\ -3 \end{pmatrix}+\begin{pmatrix} -2 \\ -1 \\ 1 \end{pmatrix}$$

$$=\begin{pmatrix} 12 \\ -1 \\ 21 \end{pmatrix}+\begin{pmatrix} -2 \\ -1 \\ 1 \end{pmatrix}=\begin{pmatrix} 12-2 \\ -1 -1 \\ 21+1 \end{pmatrix}=\begin{pmatrix} 10 \\ -2 \\ 22 \end{pmatrix}$$

$$a^T\cdot b=\begin{pmatrix} 1 & 2 & 3 \end{pmatrix}\cdot\begin{pmatrix} -2 \\ 1 \\ -3 \end{pmatrix}$$

$$=1(-2)+2(1)+3(-3)=-9$$