Question

Find the coordinates of the point which divides the line segment joining
the points (1, –2, 3) and (3, 4, –5) in the ratio 2 : 3 (i) internally, and (ii) externall

Answer 1

@terenzreignz

Answer 2

What do internally and externally mean?

Answer 3

Probably able to use vectors again... find the vector from one point to the other, then multiply it by 2/5 or 3/5 (guessing that's internally/externally), then find it's endpoint.

Answer 4

if internally use +sign ..externally -ve sign

Answer 5

If the points are A and B, find the endpoint by adding 2/5 of the vector AB, to point A

Answer 6

Possibly similar to the midpoint formula, which goes
\[\huge \left(\frac12(x_1+x_2),\frac12(y_1+y_2)\frac12(z_1+z_2)\right)\]
Except instead of \(\large \frac12\), you multiply \(\large \frac25\) instead.

Answer 7

^that looks like it should work too (looks kinda like the result of what i said, just from trying to do it in my head)

Answer 8

OH so we can also solve it using determinant or what ?

Answer 9

Determinants? This looks too simple to have to resort to determinants... :/

Answer 10

oh

Answer 11

On second thought, that formula above may be too good to be true...

Answer 12

With vectors I think it'd be \[\large (x _{1}+\frac{ 2 }{ 5 }(x _{2}-x _{1}), y _{1}+\frac{ 2 }{ 5 }(y _{2}-y _{1}), z _{1}+\frac{ 2 }{ 5 }(z _{2}-z _{1}))\]

Hopefully that's right, I'm too lazy to check on paper. It doesn't look exactly like @terenzreignz formula...

Answer 13

Yeah... that's more like... It's just that when you take \(\frac12\) it should result into the more nice-looking midpoint formula.

Yikes... I'm losing my touch :D

Answer 14

Yeah I assumed yours would work too... but it only works with half because 1-1/2 = 1/2

Answer 15

Yup :)
I learned.

Answer 16

Me too :) i figured that might be a handy shortcut for finding points not halfway between two points.

Answer 17

I guess externally, like @AravindG said, would just be subtracting 2/5. It doesn't seem to specify from which point to start from, though, and there's multiple possible points depending on which point you take the 2:5 ratio from.

Answer 18

@terenzreignz and @goformit100 there is actually a formula for this question (it's essentially what i posted above):

http://www.learnnext.com/lesson/CBSE-X-Maths-Section-Formula.htm

Answer 19

It only gives for 2 dimensions but you just need to add the third in the same way.