Question

Find the coordinate of the point R which divide the joint of the points P (0,0,0) and Q(4,-1,-2) in the ratio 1:2 externally and verify that P is the midpoint of RQ.
Please help!

Answer 1

Answer: Showed.

Step-by-step explanation: Given that the point R divides the joint of the points P(0, 0, 0) and Q(4, -1, -2) in the ratio 1 : 2 externally.

We know that if a point divides the joint of the two points (a, b, c) and (d, e, f) in the ratio m : n externally, then its co-ordinates are

$\left(\dfrac{md-na}{m-n},\dfrac{me-nb}{m-n},\dfrac{mf-nc}{m-n}\right).$

Here, m : n = 1 : 2. So, the co-ordinates of point R will be

]$\left(\dfrac{1\times 4-2\times 0}{1-2},\dfrac{1\times (-1)-2\times 0}{1-2},\dfrac{1\times (-2)-2\times 0}{1-2}\right)\\\\\\=\left(\dfrac{4}{-1},\dfrac{-1}{-1},\dfrac{-2}{-1}\right)\\\\\\=(-4,1,2).$.

Also, the co-ordinates of the mid-point of RQ are

$\left(\dfrac{-4+4}{2},\dfrac{1-1}{2},\dfrac{2-2}{2}\right)=(0,0,0),$

which are the co-ordinates of the point P.

So, the point P is the mid-point of RQ.

Hence proved.

Answer 2

$\mathfrak{Answer}$

Given :-

The co-ordinates of P(0,0,0)

The co-ordinates of Q(4,-1,-2)

Ratio which R divides the join of P and Q is -1:2

Formula of external division :-

$(mx_{2} - nx_{1}) / (m-n ), (my_{2} - ny_{1}) / (m-n ), (mz_{2} - nz_{1}) / (m-n)$

Formula of mid-point :-

$x+y / 2 , y+z/2$

Solution :-

m = 1

n = 2

$x_{1} = 0\\x_{2} = 4\\y_{1} = 0\\y_{2} = -1\\z_{1} = 0\\z_{2} = -2$

Now put the values accordingly on the formula of external division .

Answer = (-4,1,2)

Now to verify ,

$4+(-4)/2 = 0 \\-1 + 1 / 2 = 0 \\-2 + 2 / 2 = 0$

 (Coordinates of Q + Coordinates of R) / 2 = Coordinates of P (0,0,0)

Verified