Chapter 13.3, Problem 51E

(a)

To determine

To explain: Whether the given statement “ ${\text{proj}}_{v}u={\text{proj}}_{u}v$“ is true or false.

If false, give a counterexample.

(b)

To determine

To explain: Whether the given statement “If nonzero vectors u and v have the same magnitude they make equal angles with $u+v$“ is true or false. If false, give a counterexample.

(c)

To determine

To explain: Whether the given statement “ ${\left(u\cdot i\right)}^2+{\left(u\cdot j\right)}^2+{\left(u\cdot k\right)}^2={\left|u\right|}^2$“ is true or false. If false, give a counterexample.

(d)

To determine

To explain: Whether the given statement “If $u$ is orthogonal to $v$ and $v$ is orthogonal to $w$, then $u$ is orthogonal to $w$.“ is true or false. If false, give a counterexample.

(e)

To determine

To explain: Whether the given statement “The vectors orthogonal to $\langle 1,1,1\rangle$ lie on the same line.“ is true or false. If false, give a counterexample.

(f)

To determine

To explain: Whether the given statement “If the ${\text{proj}}_{v}u=0$,then u and v (both nonzero) are orthogonal.“ is true or false. If false, give a counterexample.