A sum of scalar multiple of two or more vectors (such as c_{1} u + c_{2} v + c_{3} w, where c_{i} are scalars) is called a linear combination of the vectors. Express \langle 4, -8 \rangle as a linear combination of i and j

beljuA 2021-08-08 Answered
A sum of scalar multiple of two or more vectors (such as \(\displaystyle{c}_{{{1}}}{u}+{c}_{{{2}}}{v}+{c}_{{{3}}}{w}\), where \(\displaystyle{c}_{{{i}}}\) are scalars) is called a linear combination of the vectors. Let \(\displaystyle{i}={\left\langle{1},{0}\right\rangle},{j}={\left\langle{0},{1}\right\rangle},{u}={\left\langle{1},{1}\right\rangle}\), and \(\displaystyle{v}={\left\langle-{1},{1}\right\rangle}\).
Express \(\displaystyle{\left\langle{4},-{8}\right\rangle}\) as a linear combination of i and j (that is, find scalars \(\displaystyle{c}_{{{1}}}\) and \(\displaystyle{c}_{{{2}}}\) such that \(\displaystyle{\left\langle{4},-{8}\right\rangle}={c}_{{{1}}}{i}+{c}_{{{2}}}{j}\)).

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Expert Answer

Cristiano Sears
Answered 2021-08-09 Author has 18405 answers
Step 1
Given
\(\displaystyle{i}={\left\langle{1},{0}\right\rangle}\) and \(\displaystyle{j}={\left\langle{0},{1}\right\rangle}\)
Step 2
\(\displaystyle{\left\langle{4},-{8}\right\rangle}={4}{i}-{8}{j}\)
so \(\displaystyle{c}_{{{1}}}={4}\) and \(\displaystyle{c}_{{{2}}}=-{8}\)
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