The gradient vector gradf$(a,b)$ at a point $P=(a,b)$ and several unit vectors vec$(r),$vec$(s),$vec$(t),$vec$(u),$vec$(n),$ and vec$(e)$ in the xy$-$plane are shown in the figure. You may assume that angles that look the same are the same. Here, $f_{\frac{?}{\text{b}\text{a}\text{r}}(p)}(a,b)$ denotes the directional derivative, which is often written as $D_{i}f(a,b).$

Are the following statements true or false?

:$0,0$:$)f_7(a,b)=f_i(a,b)f_a(a,b)=f_i(a,b)f_{\frac{?}{\text{b}\text{a}\text{r}}(F)}(a,b).f_F(a,b).$

$3.f_k(a,b)=||gradf(a,b)||.$

$4.f_z(a,b)$ points $in$ the direction $of$ minimum rate $of$ change $off.$

$5.f_z(a,b)=($:$0,0$:$).$

$6.f_7(a,b)=f_i(a,b).$

$7.f_a(a,b)=f_i(a,b).$

$8.f_{\frac{?}{\text{b}\text{a}\text{r}}(F)}(a,b).($Click on graph to enlarge$)$