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Worksheet 33.4 Dot products and projection

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Let \({\vec v}_1 = (-4,1)\text{,}\) \({\vec v}_2 = (2,2)\text{,}\) \({\vec v}_3 = (1,2,3)\text{,}\) \({\vec v}_4 = (-2,1,0)\text{.}\) Find the values of the following expressions:
\({\vec v}_1 \cdot {\vec v}_2 = \fillinmath{XXX}\)
\({\vec v}_3 \cdot {\vec v}_4 = \fillinmath{XXX}\)
\(\lVert{\vec v}_1\rVert = \fillinmath{XXX}\)
\(\lVert{\vec v}_4\rVert = \fillinmath{XXX}\)
Are any of these vectors perpendicular to each other?


The vectors \(\vec a = (3,9)\) and \(\vec u = (4,2)\) are pictured below. Derive the formula for projection on a line and use it to find the projection of \(\vec a\) on the line spanned by \(\vec u\text{.}\) Also compute the length of the residual vector.
two vectors in a Cartesian plane


Consider the vector equation
\begin{equation*} m \begin{bmatrix}2 \\ 5\end{bmatrix} = \begin{bmatrix}3 \\ 7\end{bmatrix}\text{.} \end{equation*}
Check that there is no solution \(m\) that makes the equation true.
Use projection to find the best approximation \(\hat m\text{.}\)
Compute \(\hat m \begin{bmatrix}2 \\ 5\end{bmatrix} \text{.}\)
Compute the length of the residual vector and explain what it means.


Consider the system of equations
\begin{align*} 3t \amp =5\\ 2t \amp = 9\text{.} \end{align*}
Find the best estimate, \(\hat t\text{,}\) of \(t\) using projection.
Compute the length of the residual vector.