Question

1. amy is pulling a wagon with a force of 30 pounds up a hill at an angle of 25°. give the force exerted on the wagon as a vector and as a linear combination of unit vectors i and j.

Explanation:

Step1: Recall vector - component formula

If a vector $\vec{F}$ has magnitude $F$ and makes an angle $\theta$ with the positive $x$ - axis, then $\vec{F}=F\cos\theta\vec{i}+F\sin\theta\vec{j}$.

Step2: Identify the values of $F$ and $\theta$

Here, $F = 30$ pounds and $\theta=25^{\circ}$.

Step3: Calculate the $x$ - component

The $x$ - component of the force vector is $F_x=F\cos\theta$. So, $F_x = 30\cos(25^{\circ})\vec{i}$. Using a calculator, $\cos(25^{\circ})\approx0.9063$, so $F_x\approx30\times0.9063\vec{i}=27.189\vec{i}$.

Step4: Calculate the $y$ - component

The $y$ - component of the force vector is $F_y=F\sin\theta$. So, $F_y = 30\sin(25^{\circ})\vec{j}$. Using a calculator, $\sin(25^{\circ})\approx0.4226$, so $F_y\approx30\times0.4226\vec{j}=12.678\vec{j}$.

Step5: Write the force vector as a linear combination

The force vector $\vec{F}$ can be written as $\vec{F}=30\cos(25^{\circ})\vec{i}+30\sin(25^{\circ})\vec{j}\approx27.19\vec{i}+12.68\vec{j}$.

Answer:

$\vec{F}\approx27.19\vec{i}+12.68\vec{j}$