Let c be the counter-clockwise planar circle with center at the origin and radius r> 0. without computing them, determine for the following vector fields\bf fwhether the line integrals\displaystyle \int_c \mathbf f \cdot \, d\mathbf{r}are positive, negative, or zero and type p, n, or z as appropriate.
a.\bf f= the radial vector field =x\mathbf i + y\mathbf j:
b.\bf f= the circulating vector field =-y\mathbf i + x\mathbf j:
c.\bf f= the circulating vector field =y\mathbf i - x\mathbf j:
d.\bf f= the constant vector field =\bf i + j:

g = (2, 2)

step-by-step explanation:

given the ratio is 1 : 1 then f is the midpoint of eg

let the coordinates of g = (x, y)

using the midpoint formula

(0 + x) = 1 ( multiply both sides by 2 )

0 + x = 2 ⇒ x = 2

similarly

(4 + y) = 3 ( multiply both sides by 2 )

4 + y = 6 ( subtract 4 from both sides )

y = 2

coordinates of g = (2, 2)

for this case we must apply distributive property to the following expression:

the distributive property states that:

then, when applying the property we have to multiply the -2 for each term within the parenthesis:

chemists are often responsible for designing a chemical reaction and analyzing the products obtained from it. one significant part of designing a chemical reaction is determining the amount of each reactant that is needed and the amount of each product that will be produced. in order to do so, chemists use stoichiometry, the quantitative relationships between reactants and products.