Now, Now,We are given the following vectors:
[tex]P\left(5,4\right),Q\left(7,3\right),R\left(8,6\right),S\left(4,1\right)[/tex]We are asked to determine the following vector:
[tex]PQ+3RS[/tex]First, we will determine the vector PQ and RS. To determine PQ we use the following:
[tex]PQ=Q-P[/tex]This means we need to subtract "P" from "Q". We do that by subtracting each component of the points, like this:
[tex]PQ=\left(7,3\right)-\lparen5,4)=\left(7-5,3-4\right)[/tex]Solving the operations:
[tex]PQ=\left(2,-1\right)[/tex]Now, we use a similar procedure to determine RS:
[tex]RS=S-R[/tex]Substituting we get:
[tex]RS=\left(4,1\right)-\left(8,6\right)=\left(4-8,1-6\right)[/tex]Solving the operations:
[tex]RS=\left(-4,-5\right)[/tex]Now, we substitute the values in the vector we are looking for:
[tex]PQ+3RS=\left(2,-1\right)+3\left(-4,-5\right)[/tex]Now, we solve the product by multiplying both components of RS:
[tex]PQ+3RS=(2,-1)+(-12,-15)[/tex]Now, we solve the addition by adding each corresponding component:
[tex]PQ+3RS=(2-12,-1-15)[/tex]Solving the operations:
[tex]PQ+3RS=(-10,-16)[/tex]And thus we have determined the components.
Part B. We area asked to determine the magnitude of the vector. To do that we will use the following:
Given a vector of the form:
[tex]X=\left(x,y\right)[/tex]Its magnitude is:
[tex]\lvert X\rvert=\sqrt{x^2+y^2}[/tex]This means that the magnitude is the square root of the sum of the square of the components. Applying the formula we get:
[tex]\lvert\begin{equation*}PQ+3RS\end{equation*}\rvert=\sqrt{\left(-10\right)^2+\left(-16\right)^2}[/tex]Now, we solve the squares:
[tex]\lvert\begin{equation*}PQ+3RS\end{equation*}\rvert=\sqrt{100+256}[/tex]Solving the addition:
[tex]\lvert\begin{equation*}PQ+3RS\end{equation*}\rvert=\sqrt{356}[/tex]Now, we factor the term inside the radical as follows:
[tex]\lvert PQ+3RS\rvert=\sqrt{4\left(89\right)}[/tex]Now, we distribute the radical:
[tex]\lvert PQ+3RS\rvert=\sqrt{4}\sqrt{89}[/tex]Taking the left square root:
[tex]\lvert PQ+3RS\rvert=2\sqrt{89}[/tex]And thus we have determined the magnitude.
