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Given AB=(-4) and CA=(3/5), find BC.

October 10, 2026Mathematics

ScanSolve AI Answer

Answer

\begin{pmatrix} 1 \\ -2 \end{pmatrix}

5 steps

Identify the given vectors.
We are given the vectors $\vec{AB}$ and $\vec{CA}$ .
$\vec{AB} = \begin{pmatrix} -4 \\ 7 \end{pmatrix}$ $\vec{CA} = \begin{pmatrix} 3 \\ -5 \end{pmatrix}$
Express $\vec{BC}$ in terms of the given vectors.
Using the triangle rule for vector addition, $\vec{BC} = \vec{BA} + \vec{AC}$ .
We know that $\vec{BA} = -\vec{AB}$ and $\vec{AC} = -\vec{CA}$ .
Calculate $\vec{BA}$ .
$\vec{BA} = -\vec{AB} = -\begin{pmatrix} -4 \\ 7 \end{pmatrix} = \begin{pmatrix} -(-4) \\ -(7) \end{pmatrix} = \begin{pmatrix} 4 \\ -7 \end{pmatrix}$
Calculate $\vec{AC}$ .
$\vec{AC} = -\vec{CA} = -\begin{pmatrix} 3 \\ -5 \end{pmatrix} = \begin{pmatrix} -(3) \\ -(-5) \end{pmatrix} = \begin{pmatrix} -3 \\ 5 \end{pmatrix}$
Calculate $\vec{BC}$ .
Now, substitute the values of $\vec{BA}$ and $\vec{AC}$ into the expression for $\vec{BC}$ :
$\vec{BC} = \vec{BA} + \vec{AC} = \begin{pmatrix} 4 \\ -7 \end{pmatrix} + \begin{pmatrix} -3 \\ 5 \end{pmatrix}$ $\vec{BC} = \begin{pmatrix} 4 + (-3) \\ -7 + 5 \end{pmatrix}$ $\vec{BC} = \begin{pmatrix} 1 \\ -2 \end{pmatrix}$
The final answer is $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$ .

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