Activity 3.8.1.
Let’s look at the vector \(\vec{w}=\colvec{7 \\-2}\) as a vector in \(\mathbb{R}^2\text{.}\)
(a)
How much does \(\vec{w}=\colvec{7 \\-2}\) move horizontally?
(b)
How much does \(\vec{w}=\colvec{7 \\-2}\) move vertically?
(c)
Calculate the dot product of \(\vec{w}=\colvec{7 \\-2}\) with \(\colvec{1\\0}\text{.}\) How does this relate to your previous answers?
(d)
Calculate the dot product of \(\vec{w}=\colvec{7 \\-2}\) with \(\colvec{0\\1}\text{.}\) How does this relate to your previous answers?
(e)
How can you write \(\vec{w}=\colvec{7 \\-2}\) as a linear combination of \(\colvec{1\\0}\) and \(\colvec{0\\1}\text{?}\)
(f)
How much does \(\vec{w}=\colvec{7 \\-2}\) move in the \(\colvec{1\\1}\) direction?
(g)
How much does \(\vec{w}=\colvec{7 \\-2}\) move in the \(\colvec{-1\\1}\) direction?
(h)
Calculate the dot product of \(\vec{w}=\colvec{7 \\-2}\) with \(\colvec{1\\1}\text{.}\) How does this relate to your previous answers?
(i)
Calculate the dot product of \(\vec{w}=\colvec{7 \\-2}\) with \(\colvec{-1\\1}\text{.}\) How does this relate to your previous answers?
(j)
How can you write \(\vec{w}=\colvec{7 \\-2}\) as a linear combination of \(\colvec{1\\1}\) and \(\colvec{-1\\1}\text{?}\)
(k)
How much does \(\vec{w}=\colvec{7 \\-2}\) move in the \(\colvec{1\\1}\) direction?
(l)
How much does \(\vec{w}=\colvec{7 \\-2}\) move in the \(\colvec{1\\0}\) direction?
(m)
Calculate the dot product of \(\vec{w}=\colvec{7 \\-2}\) with \(\colvec{1\\1}\text{.}\) How does this relate to your previous answers?
(n)
Calculate the dot product of \(\vec{w}=\colvec{7 \\-2}\) with \(\colvec{1\\0}\text{.}\) How does this relate to your previous answers?
(o)
How can you write \(\vec{w}=\colvec{7 \\-2}\) as a linear combination of \(\colvec{1\\1}\) and \(\colvec{1\\0}\text{?}\)
(p)
What is different about the different sets we considered to span \(\mathbb{R}^2\) in this activity? Be specific about what aspects will make answering the linear combination question easier to answer.
