Question

F1=110 cos 50°i + 110 sin 50°j F2=60 cos 160°i + 60 sin 160°jPart A - Find their dot productPart B - Use the dot product to find the angle between them

Answer 1

ANSWER :

Part A :

EXPLANATION :

Part A :

Note that the dot product of two vectors is given by :

$\begin{gathered} A=ai+bj\quad and\quad B=ci+dj \\ A\cdot B=a(c)+b(d) \end{gathered}$

From the problem, we have the vectors :

$\begin{gathered} F_1=110\cos50i+110\sin50j \\ F_2=60\cos160i+60\sin160j \end{gathered}$

The dot product will be :

$\begin{gathered} F_1\cdot F_2=110\cos50(60\cos160)+110\sin50(60\sin160) \\ =-3986.55+1729.22 \\ =-2257.33 \end{gathered}$

Part B :

The cosine of the angle between two vectors is given by :

$\cos\theta=\frac{F_1\cdot F_2}{\lvert{F_1}\rvert\lvert{F_2}\rvert}$

Solve for the |F1| and |F2|

$\begin{gathered} \lvert{F_1}\rvert=\sqrt{(110\cos50)^2+(110\sin50)^2}=110 \\ \lvert{F_2}\rvert=\sqrt{(60\cos160)^2+(60\sin160)^2}=60 \end{gathered}$

Now substitute the given values :

$\begin{gathered} \cos\theta=\frac{-2257.33}{110(60)} \\ \text{ Using arccosine :} \\ \arccos(\cos\theta)=\arccos(\frac{-2257.33}{110\times60}) \\ \theta=110 \end{gathered}$

The angle between two vectors is 110 degrees