What is the area of a triangle with vertices P(2,3,5), Q(4,2,-1), R(3,6,4)?

Answer 1

3.5 sq

Answer 2

I think you will need the formula 16A^2=(a^2+b^2+c^2)^2-2(a^4+b^4+c^4). That's pretty complicated but I couldn't find an easier way.
So, first of all you need to find a, b, and c. Let's say a is the distance between P and Q.
So you need the formula p-q=sqrt((P1-Q1)^2+(P2-Q2)^2+(P3-Q3)^2).
That gives you b=sqrt((2-4)^2+(3-2)^2+(5+1)^2)=sqrt41=6.403
So now you got to find b and c.
b=q-r=sqrt((4-3)^2+(2-6)^2+(-1-4)^2)=sqrt42=6.481
c=p-r=sqrt((2-3)^2+(3-6)^2+(5-4)^2)=sqrt11=3.317
So now you put those numbers in the formula:
16A^2=(41+42+11)^2-2(41^2+42^2+11^2)
16A^2=94^2-3566
16A^2=8836-3566
16A^2=5270
A^2=329.375
A=18.1487
It seems like there should be anm easier way to do this, but I hope it's right :-)

Answer 3

area is 1/2 mod(A xB)

if you say A = P-Q so

(2i+3j+5k) -(4i + 2j - k)

a= -2i + j + 6K

and then the same with p-r to get b

ill assume you know how to do the vector product and as i cant be bothered to try and lay it out on this ill skip straight to the next part

i got 19i - 4j+7K and the mod of this is root 426 hence the area is 1/2 root 426 or 10.32

Answer 4

como got the right answer-
now here is a different way:-
P-Q=(-2,1,6)
Q-R=(1,-4,-5)
R-P=(1,3,-1)
{note that the sum is (0,0,0)}
a=sqrt(4+1+36)=sqrt41
b=sqrt(1+16+25)=sqrt42
c=sqrt(1+9+1)=sqrt11
s=(sqrt41+sqrt42+sqrt11)/2
(s-a)
=(sqrt42+sqrt11-sqrt41)/2
(s-b)
=(sqrt41+sqrt11-sqrt42)/2
(s-c)
=(sqrt41+sqrt42-sqrt11)/2
area of triangle K
=sqrt{s(s-a)(s-b)(s-c)}

i substituted the above values
into the equation and after
a bit of algebra got;
K=sqrt[{2*sqrt41*sqrt42-72}
{2*sqrt41*sqrt42+72}]/4
=sqrt{4*41*42-72^2}/4
=sqrt{6888-5184}/4
=sqrt{1704}/4
=sqrt{4*426}/4
=(1/2)sqrt(426)
=10.31988372 sq units

i hope that this helps

Answer 5

Vector PQ = 2i - j - 6k
Vector PR = i + 3j - k

Area of Triangle = 1/2 DET of vector PQ cross vector PR

= 1/2 | i--------j--------- k |
--------|2----- -1------- -6 |
--------|1-------3--------1 |

=1/2|19i - 4j + 7k|

= (1/2).√(19² + (- 4)² + 7²)

= (1/2). √ 426

PS This is guaranteed correct as I have checked answer against answer page of text book!

Answer 6

3.5 sq

Answer 7

3.5sq?maybe