Let $F$ be a constant unit force that is parallel to the vector $(1, 0, 1)$ in xyz-space. What is the work done by $F$ on a particle that moves along the path given by $(t, t^2, t^3)$ between time $t=0$ and time $t=1$?
Let $f$ be real valued function defined and continuous on the set of real numbers $R$. Is it true that the set $S=\{f(c): 0<c<1\}$ is a bounded subset of $R$?
Let $V$ be the real vector space of all real $2\times 3$ matrices and let $W$ be the real vector space of all real $4\times 1$ column vectors. If $T$ is a linear transformation from $V$ onto $W$. What is the dimension of the subspace $\{v\in V: T(v)=0\}$?
Thanks in advance.
Hints: $1$) The only thing that matters is the component of $(1,1,1)$ in the $(1,0,1)$ direction.
$2$) Consider the function on $[0,1]$ and quote a standard theorem.
$3$) The vector space $V$ has dimension $6$.