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Worksheet 34.4 Dot products and projection
1. 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:
(a) \({\vec v}_1 \cdot {\vec v}_2 = \fillinmath{XXX}\)
(b) \({\vec v}_3 \cdot {\vec v}_4 = \fillinmath{XXX}\)
(c) \(\lVert{\vec v}_1\rVert = \fillinmath{XXX}\)
(d) \(\lVert{\vec v}_4\rVert = \fillinmath{XXX}\)
(e) Are any of these vectors perpendicular to each other?
2. 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.
3.
Consider the vector equation
\begin{equation*}
m \begin{bmatrix}2 \\ 5\end{bmatrix} = \begin{bmatrix}3 \\ 7\end{bmatrix}\text{.}
\end{equation*}
(a) Check that there is no solution \(m\) that makes the equation true.
(b) Use projection to find the best approximation \(\hat m\text{.}\)
(c) Compute \(\hat m \begin{bmatrix}2 \\ 5\end{bmatrix} \text{.}\)
(d) Compute the residual vector.
(e) Compute the length of the residual vector and explain what it means.
4.
Consider the system of equations
\begin{align*}
3t \amp =5\\
2t \amp = 9\text{.}
\end{align*}
(a) Write the system in vector form.
(b) Find the best estimate, \(\hat t\text{,}\) of \(t\) using projection.
(c) Compute the length of the residual vector.
