• :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon Suggested languages for you:

Europe

Answers without the blur. Sign up and see all textbooks for free! Q. 56

Expert-verified Found in: Page 603 ### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465 # If $P=\left(-3,1\right)andQ=\left(x,4\right)$, find all numbers x such that the vector represented by $\stackrel{⇀}{PQ}$ has length 5.

There will be two values for x which are -7 and 1.

See the step by step solution

## Step. 1 Given

We have coordinates of two points,

$P\equiv \left(-3,1\right)andQ\equiv \left(x,4\right)$

and, length of the vector

$\stackrel{⇀}{PQ}is5$

i.e.,

$\left|\stackrel{⇀}{PQ}\right|=5$

## Step. 2 Formula to find length of a vector

As we all know,

The length of the vector having coordinates $\left({x}_{1},{y}_{1}\right)and\left({x}_{2},{y}_{2}\right)$ is

$\sqrt{{\left({x}_{1}-{x}_{2}\right)}^{2}+{\left({y}_{1}-{y}_{2}\right)}^{2}}units$

Now, $\stackrel{⇀}{PQ}=\left(x-\left(-3\right)\right)\stackrel{⇀}{i}+\left(4-1\right)\stackrel{⇀}{j}\phantom{\rule{0ex}{0ex}}\stackrel{⇀}{PQ}=\left(x+3\right)\stackrel{⇀}{i}+3\stackrel{⇀}{j}$,

## Step. 3 Calculating the value of x

Now we have,

$\left|\stackrel{⇀}{PQ}\right|=5\phantom{\rule{0ex}{0ex}}\sqrt{{\left(x+3\right)}^{2}+{\left(3\right)}^{2}}=5$,

Squaring both sides we have,

${\left(x+3\right)}^{2}+9=25,\phantom{\rule{0ex}{0ex}}{\left(x+3\right)}^{2}=16,$

Now taking square root both sides we get,

$\left(x+3\right)=±4,\phantom{\rule{0ex}{0ex}}So,\phantom{\rule{0ex}{0ex}}x=±4-3,\phantom{\rule{0ex}{0ex}}i.e.,\phantom{\rule{0ex}{0ex}}x=1or-7$ ### Want to see more solutions like these? 