Answer to Question #148578 in Analytic Geometry for Jyotiramay Rout

Question #148578
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,
1
Expert's answer
2020-12-04T12:52:08-0500
a=(123),b=(213),c=(211)a=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, b=\begin{pmatrix} -2 \\ 1 \\ -3 \end{pmatrix}, c=\begin{pmatrix} -2 \\ -1 \\ 1 \end{pmatrix}

2a5b=2(123)5(213)=(2(1)5(2)2(2)5(1)2(3)5(3))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}

=(12121)=\begin{pmatrix} 12 \\ -1 \\ 21 \end{pmatrix}

2a5b+c=2(123)5(213)+(211)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}

=(12121)+(211)=(1221121+1)=(10222)=\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}

aTb=(123)(213)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=1(-2)+2(1)+3(-3)=-9


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