Question

write the vector in the form ⟨a,b⟩.
the vector is ⟨□,□⟩.
(simplify your answer. do not round until the final answer. then round to four decimal places as needed.)

Explanation:

Step1: Recall vector component formula

For a vector with magnitude \( r \) and angle \( \theta \) (from positive x - axis), the components are \( a = r\cos\theta \) and \( b = r\sin\theta \). Here, \( r = 4 \) and \( \theta=140^{\circ} \).

Step2: Calculate the x - component (\( a \))

\( a = 4\cos(140^{\circ}) \). We know that \( \cos(140^{\circ})=\cos(180^{\circ} - 40^{\circ})=-\cos(40^{\circ}) \). So \( a = 4\times(-\cos(40^{\circ}))\approx4\times(- 0.7660444431)=-3.064177772 \).

Step3: Calculate the y - component (\( b \))

\( b = 4\sin(140^{\circ}) \). We know that \( \sin(140^{\circ})=\sin(180^{\circ}-40^{\circ})=\sin(40^{\circ}) \). So \( b = 4\times\sin(40^{\circ})\approx4\times0.6427876097 = 2.571150439 \).

Step4: Round to four decimal places

For \( a \), rounding \( - 3.064177772 \) to four decimal places gives \( - 3.0642 \). For \( b \), rounding \( 2.571150439 \) to four decimal places gives \( 2.5712 \).

Answer:

\(\langle - 3.0642,2.5712
angle\)