The Dot Product & Projections
Exercises
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For the vectors
\(\bm{u} = \langle 1,3 \rangle\) and \(\bm{v} = \langle -4,2 \rangle,\)
calculate \(\bm{u}\cdot\bm{v}.\)
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For the vectors
\(\bm{u} = \langle 5,4 \rangle\) and \(\bm{v} = \langle 1,-3 \rangle,\)
calculate \(\bm{u}\cdot\bm{v}.\)
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What is the angle between the vectors
\(\bm{u} = \langle 1,3 \rangle\) and \(\bm{v} = \langle -4,2 \rangle?\)
What is the area of the triangle framed by those vectors?
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What is the angle between the vectors
\(\bm{u} = \langle 5,4 \rangle\) and \(\bm{v} = \langle 1,-3 \rangle?\)
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What is the area of the triangle in the plane
with vertices located at the coordinates
\((1,1)\) and \((3,-3)\) and \((2,7)?\)
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Suppose that \(\bm{u} = \bigl\langle 3,-5 \bigr\rangle\)
and that \(\bm{v}\) is some unknown vector.
Is it possible that
\(\operatorname{proj}_{\bm{u}}(\bm{v}) = \bigl\langle 2,-3\bigr\rangle?\)
If it is, determine what \(\bm{v}\) is.
If not, why not?
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Make a sketch of the vectors
\(\bm{u} = \langle 1,3 \rangle\) and \(\bm{v} = \langle -4,2 \rangle.\)
Then on your sketch add in vectors for
the projection of \(\bm{u}\) onto \(\bm{v}\) (\(\operatorname{proj}_{\bm{v}}(\bm{u})\))
and the projection of \(\bm{v}\) onto \(\bm{u}\) (\(\operatorname{proj}_{\bm{u}}(\bm{v})\)),
and calculate the components of these vectors explicitly.
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Make a sketch of the vectors
\(\bm{u} = \langle 5,4 \rangle\) and \(\bm{v} = \langle 1,-3 \rangle.\)
Then on your sketch add in vectors for
the projection of \(\bm{u}\) onto \(\bm{v}\) (\(\operatorname{proj}_{\bm{v}}(\bm{u})\))
and the projection of \(\bm{v}\) onto \(\bm{u}\) (\(\operatorname{proj}_{\bm{u}}(\bm{v})\)),
and calculate the components of these vectors explicitly.
Problems & Challenges
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If \(\bm{v} = \langle 3,2 \rangle\)
and \(\bm{u}\cdot\bm{v} = 5\)
and the angle between \(\bm{u}\) and \(\bm{v}\) is 30°,
what must \(|\bm{u}|\) be?
Explicitly, in terms of its components,
what vector might \(\bm{u}\) be?
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Suppose that \(\bm{u} = \bigl\langle 5,7 \bigr\rangle\)
and that the \(\bigl|\operatorname{proj}_{\bm{u}}(\bm{v})\bigr| = 3.\)
Write down three different possibilities,
expressed explicitly in terms of its components,
for the vector \(\bm{v}.\)