Derivatives and Integrals: An Annotated Discourse

Worksheet33.4Dot products and projection A4US

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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.