Question

Short answer

The vector field \(F\) matches with the plot II.

Step-by-step solution

Given Information.

The given vector field is\(F\left( {x,y} \right) = \left\langle {\cos \left( {x + y} \right),x} \right\rangle \).

Explanation.

Vector fields generally have components in the \(i,{\rm{ }}j,{\rm{ and }}k\)axis in order to present the direction and magnitude of a group of vectors (vector fields).

These components can be either constants or a function of variables\(x,{\rm{ }}y,{\rm{ or }}z\).

\(\begin{aligned}{l}F\left( {x,y} \right) &= \left\langle {\cos \left( {x + y} \right),x} \right\rangle \\ \Rightarrow F\left( {x,y} \right) &= \cos \left( {x + y} \right)i + xj\end{aligned}\)

The vector field has components as functions of variable either\(x{\rm{ or }}y\). That means, the slope and the magnitude of each of the vector will be depending on the coordinates\(\left( {x,y} \right)\).

Find slope and magnitude.

To find the slope and the magnitude:

Let,

\(\begin{aligned}{l}F(x,y) &= Pi + Qj\\ \Rightarrow F\left( {x,y} \right) &= \cos \left( {x + y} \right)i + xj\end{aligned}\)

Then,

\(\begin{aligned}{l}P &= \cos \left( {x + y} \right)\\Q &= x\end{aligned}\)

The slope will be,

\(\frac{Q}{P} = \frac{x}{{\cos \left( {x + y} \right)}}\)

The magnitude will be,

\(\begin{aligned}{l}\sqrt {{P^2} + {Q^2}} \\ &= \sqrt {{x^2} + co{s^2}\left( {x + y} \right)} \end{aligned}\)

Analyse slope and magnitude.

Let \(y\)be positive constant, as \(x\) increases from \(x = y\), the slope goes to zero to infinity and the change of the slope gradually increases.

For the same value of \(y\), as \(x\) increases from \(x = y\), the magnitude increases to infinity.

Let \(x\) be positive constant, as \(y\) increases from \(x = y\), the slope goes from zero to infinity and then goes back to zero i.e., vector turns \(360\)degree.

For the same value of \(x\), as \(y\) increases from\(x = y\), the magnitude increases to a certain point and then decreases back to the original length.

Therefore, the vector field \(F\) matches with the plot II.