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Consider the vectors x1 = (8, 6)T and x2 = (4, 1)T in R2. (a) Determine the length of
Chapter 3, Problem 1(choose chapter or problem)
Consider the vectors \(\mathbf{x}_{1}=(8,6)^{T}\) and \(\mathbf{x}_{2}=(4,-1)^{T} \text { in } \mathbb{R}^{2}\).
(a) Determine the length of each vector.
(b) Let \(\mathbf{x}_{3}=\mathbf{x}_{1}+\mathbf{x}_{2}\). Determine the length of \(\mathbf{x}_{3}\). How does this length compare with the sum of the lengths of \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\)?
(c) Draw a graph illustrating how \(\mathbf{x}_{3}\) can be constructed geometrically using \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\). Use this graph to give a geometrical interpretation of your answer to the question in part (b).
Questions & Answers
QUESTION:
Consider the vectors \(\mathbf{x}_{1}=(8,6)^{T}\) and \(\mathbf{x}_{2}=(4,-1)^{T} \text { in } \mathbb{R}^{2}\).
(a) Determine the length of each vector.
(b) Let \(\mathbf{x}_{3}=\mathbf{x}_{1}+\mathbf{x}_{2}\). Determine the length of \(\mathbf{x}_{3}\). How does this length compare with the sum of the lengths of \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\)?
(c) Draw a graph illustrating how \(\mathbf{x}_{3}\) can be constructed geometrically using \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\). Use this graph to give a geometrical interpretation of your answer to the question in part (b).
ANSWER:Step 1 of 3
a) Since \(R^{2}\) is Euclidean vector space, lengths of vectors \(\mathbf{x}_{1}=(8,6)^{T}\) and \(\mathbf{x}_{2}=(4,-1)^{T}\), are, respectively, given by
\(\begin{array}{l}
\sqrt{8^{2}+6^{2}}=\sqrt{64+36} \\
=\sqrt{100}=10 \\
\sqrt{4^{2}+(-1)^{2}}=\sqrt{16+1}=\sqrt{17}
\end{array}\)
Length of \(\mathbf{x}_{1}\) is 10 and the length of \(\mathbf{x}_{2}\) is \(\sqrt{17}\).