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(1) If a = (3/4), b = (6/5) and c = (6/-4) find a - 2b + c.

August 4, 2026Mathematics

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Answer

\begin{pmatrix} -3 \\ -10 \end{pmatrix}

4 steps

Identify the given vectors.
We are given the vectors:
$a = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ $b = \begin{pmatrix} 6 \\ 5 \end{pmatrix}$ $c = \begin{pmatrix} 6 \\ -4 \end{pmatrix}$
Calculate $2b$ by multiplying each component of vector $b$ by 2.
$2b = 2 \times \begin{pmatrix} 6 \\ 5 \end{pmatrix} = \begin{pmatrix} 2 \times 6 \\ 2 \times 5 \end{pmatrix} = \begin{pmatrix} 12 \\ 10 \end{pmatrix}$
Substitute the vectors into the expression $a - 2b + c$ .
$a - 2b + c = \begin{pmatrix} 3 \\ 4 \end{pmatrix} - \begin{pmatrix} 12 \\ 10 \end{pmatrix} + \begin{pmatrix} 6 \\ -4 \end{pmatrix}$
Perform the subtraction and addition component-wise.
First, subtract $2b$ from $a$ :
$\begin{pmatrix} 3 \\ 4 \end{pmatrix} - \begin{pmatrix} 12 \\ 10 \end{pmatrix} = \begin{pmatrix} 3 - 12 \\ 4 - 10 \end{pmatrix} = \begin{pmatrix} -9 \\ -6 \end{pmatrix}$
Next, add $c$ to the result:
$\begin{pmatrix} -9 \\ -6 \end{pmatrix} + \begin{pmatrix} 6 \\ -4 \end{pmatrix} = \begin{pmatrix} -9 + 6 \\ -6 + (-4) \end{pmatrix} = \begin{pmatrix} -3 \\ -10 \end{pmatrix}$
The final result is:
$\begin{pmatrix} -3 \\ -10 \end{pmatrix}$

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