Arithmetic of Column Vectors: Subtraction and Resultants
Arithmetic of Column Vectors: Subtraction and Resultants
TGM Original
Square grid with x and y axes
Given that column vectors \(\mathbf{a} = \begin{pmatrix} -3 \\ 7 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 5 \\ -2 \end{pmatrix}\).
- Draw, on the \(1\) unit by \(1\) unit square grid provided, the relationship between \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{a} - \mathbf{b}\).
- Find the column vector of \(\mathbf{a} - \mathbf{b}\) numerically.
- Find the value of \(|\mathbf{a}|\), \(|\mathbf{b}|\) and \(|\mathbf{a} - \mathbf{b}|\).
- Is \(|\mathbf{a} - \mathbf{b}| = |\mathbf{a}| - |\mathbf{b}|\)? Explain your answer.
Solution:
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Answer:(i) Head-to-tail construction with \(-\mathbf b\). (ii) \(\begin{pmatrix}-8\\9\end{pmatrix}\). (iii) \(\lvert\mathbf a\rvert\approx7.62\), \(\lvert\mathbf b\rvert\approx5.39\), \(\lvert\mathbf a-\mathbf b\rvert\approx12.0\) units. (iv) No.
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