Activity 24.
First complete the problems in the exploration task. Then write your conclusion to the question in the summarize task.
(a)
Use these vectors. \(\vec{u}=(1,3,-2),\) \(\vec{v}=(8,-2,1),\) \(\vec{x}=(10,4,-3),\) \(\vec{y}=(17,-1,0).\) \(\vec{a}=(27,3,-3),\) \(\vec{b}=(-7,5,-3).\)
(i)
High noon
(A)
Calculate \(\vec{u} \cdot \vec{v}.\)
(B)
What does this tell you about the two pairs of vectors?
(C)
Calculate \(\mbox{proj}_{\vec{u}} \vec{a}\)
(D)
Calculate \(\mbox{proj}_{\vec{v}} \vec{a}\)
(E)
Calculate \(\mbox{proj}_{\vec{u}} \vec{b}\)
(F)
Calculate \(\mbox{proj}_{\vec{v}} \vec{b}\)
(G)
Find \(s,t\) such that \(\vec{a}=s \vec{u}+t \vec{v}.\)
(H)
Find \(s,t\) such that \(\vec{b}=s \vec{u}+t \vec{v}.\)
(I)
What did you notice?
(ii)
Alaskan shadows
(A)
Calculate \(\vec{x} \cdot \vec{y}.\)
(B)
What does this tell you about the two pairs of vectors?
(C)
Calculate \(\mbox{proj}_{\vec{x}} \vec{a}\)
(D)
Calculate \(\mbox{proj}_{\vec{y}} \vec{a}\)
(E)
Calculate \(\mbox{proj}_{\vec{x}} \vec{b}\)
(F)
Calculate \(\mbox{proj}_{\vec{y}} \vec{b}\)
(G)
Find \(s,t\) such that \(\vec{a}=s \vec{x}+t \vec{y}.\)
(H)
Find \(s,t\) such that \(\vec{b}=s \vec{x}+t \vec{y}.\)
(I)
What did you notice?
(J)
How did this example differ from the previous?
(iii)
Removing Shadows
(A)
Calculate \(\vec{z}=\mbox{proj}_{\vec{x}} \vec{y}\)
(B)
Calculate \(\vec{y}_0=\vec{y}-\vec{z}.\)
(C)
Calculate \(\vec{x} \cdot \vec{y}_0.\)
(D)
Calculate \(\mbox{proj}_{\vec{x}} \vec{a}\)
(E)
Calculate \(\mbox{proj}_{\vec{y}_0} \vec{a}\)
(F)
Find \(s,t\) such that \(\vec{a}=s \vec{x}+t \vec{y}_0.\)
(b)
You have re-written vectors (\(\vec{a}\) and \(\vec{b}\)) as scaled sums (called linear combinations) of a pair of other vectors (\(\vec{u},\vec{v}\) or \(\vec{x},\vec{y}\)).
(i)
Describe what you discovered about the immportance of the pair of vectors (projected onto) being orthogonal.
(ii)
Describe the effect of the process in the Removing Shadows task.
