Section 4.3 True or False?

Checkpoint 4.16 True or False.

Let \(\vec u\) and \(\vec v\) be two vectors in \(\mathbb R^3\text{.}\) Then, \(\vec u \times \text{proj}_{\vec v}\vec u\) is perpendicular to \(\vec u - \text{proj}_{\vec v}\vec u\text{.}\)

True.
Note that \(\vec u \times \text{proj}_{\vec v}\vec u\) is perpendicular to both \(\vec u\) and \(\text{proj}_{\vec v}\vec u\text{,}\) which gives:

\begin{align*} \left(\vec u \times \text{proj}_{\vec v}\vec u\right) \cdot \left(\vec u - \text{proj}_{\vec v}\vec u\right) \amp = \left(\vec u \times \text{proj}_{\vec v}\vec u\right) \cdot \vec u\\ \amp \quad - \left(\vec u \times \text{proj}_{\vec v}\vec u\right) \cdot \text{proj}_{\vec v}\vec u\\ \amp = 0 - 0 = 0, \end{align*}

and therefore \(\vec u \times \text{proj}_{\vec v}\vec u\) is perpendicular to \(\vec u - \text{proj}_{\vec v}\vec u\text{.}\)
False.
Note that \(\vec u \times \text{proj}_{\vec v}\vec u\) is perpendicular to both \(\vec u\) and \(\text{proj}_{\vec v}\vec u\text{,}\) which gives:

\begin{align*} \left(\vec u \times \text{proj}_{\vec v}\vec u\right) \cdot \left(\vec u - \text{proj}_{\vec v}\vec u\right) \amp = \left(\vec u \times \text{proj}_{\vec v}\vec u\right) \cdot \vec u\\ \amp \quad - \left(\vec u \times \text{proj}_{\vec v}\vec u\right) \cdot \text{proj}_{\vec v}\vec u\\ \amp = 0 - 0 = 0, \end{align*}

and therefore \(\vec u \times \text{proj}_{\vec v}\vec u\) is perpendicular to \(\vec u - \text{proj}_{\vec v}\vec u\text{.}\)

Checkpoint 4.17 True or False.

Let \(\vec u\) and \(\vec v\) be two standard unit vectors in \(\mathbb R^3\text{.}\) Then, \(\left(\vec u \cdot \vec v\right)^2 + \left\|\vec v \times \vec u\right\|^2 = 1\text{.}\)

True.
Note that:

\begin{align*} \left(\vec u \cdot \vec v\right)^2 + \left\|\vec v \times \vec u\right\|^2 \amp= \|\vec u\|^2\|\vec v\|^2\cos^2\theta + \|\vec u\|^2\|\vec v\|^2\sin^2\theta\\ \amp= \|\vec u\|^2\|\vec v\|^2\left(\cos^2\theta + \sin^2\theta\right) = \|\vec u\|^2\|\vec v\|^2. \end{align*}

Now, since \(\vec u\) and \(\vec v\) are standard unit vectors, we have \(\|\vec u\| = \|\vec v\| = 1\text{,}\) which means:

\begin{equation*} \left(\vec u \cdot \vec v\right)^2 + \left\|\vec v \times \vec u\right\|^2 = \|\vec u\|^2\|\vec v\|^2 = 1. \end{equation*}
False.
Note that:

\begin{align*} \left(\vec u \cdot \vec v\right)^2 + \left\|\vec v \times \vec u\right\|^2 \amp= \|\vec u\|^2\|\vec v\|^2\cos^2\theta + \|\vec u\|^2\|\vec v\|^2\sin^2\theta\\ \amp= \|\vec u\|^2\|\vec v\|^2\left(\cos^2\theta + \sin^2\theta\right) = \|\vec u\|^2\|\vec v\|^2. \end{align*}

Now, since \(\vec u\) and \(\vec v\) are standard unit vectors, we have \(\|\vec u\| = \|\vec v\| = 1\text{,}\) which means:

\begin{equation*} \left(\vec u \cdot \vec v\right)^2 + \left\|\vec v \times \vec u\right\|^2 = \|\vec u\|^2\|\vec v\|^2 = 1. \end{equation*}

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