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Worksheet 31.5 Dot products and projection
A4US

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)
\({\vec v}_1 \cdot {\vec v}_2 = \underline{\hspace{4.545454545454546em}}\)
(b)
\({\vec v}_3 \cdot {\vec v}_4 = \underline{\hspace{4.545454545454546em}}\)
(c)
\(\lVert{\vec v}_1\rVert = \underline{\hspace{4.545454545454546em}}\)
(d)
\(\lVert{\vec v}_4\rVert = \underline{\hspace{4.545454545454546em}}\)
(e)

Are any of these vectors perpendicular to each other?

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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.

two vectors...
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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.

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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.

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