Find-the-coordinate-of-the-point-in-R-3-which-is-the-reflection-the-point-1-2-3-with-respect-to-plane-X-Y-Z-1-

Question Number 17520 by 1kanika# last updated on 07/Jul/17

Find the coordinate of the point in

R Λ 3 which is the reflection the point

(1, 2, 3) with respect to plane

X + Y + Z = 1 .

Commented by 1kanika# last updated on 07/Jul/17

Here R Λ 3 means R \times R \times R

Answered by ajfour last updated on 07/Jul/17

let given point is {\bar{r}}_{0} = \hat{i} + 2 \hat{j} + 3 \hat{k}

equation of plane : \bar{r} . \bar{n} = q

if reflected point is {\bar{r}}_{1} = {\bar{r}}_{0} + 2 λ \bar{n}

then foot of perpendicular is

{\bar{r}}_{M} = {\bar{r}}_{0} + λ \bar{n}

this point is in the plane, so

{\bar{r}}_{M} . \bar{n} = q

\Rightarrow ({\bar{r}}_{0} + λ \bar{n}) . \bar{n} = q

λ = \frac{q - {\bar{r}}_{0} . \bar{n}}{\bar{n} . \bar{n}} = \frac{1 - (1 + 2 + 3)}{1 + 1 + 1} = - \frac{5}{3}

{\bar{r}}_{1} = {\bar{r}}_{0} + 2 λ \bar{n}

= (\hat{i} + 2 \hat{j} + 3 \hat{k}) - \frac{10}{3} (\hat{i} + \hat{j} + \hat{k})

= (- \frac{7}{3} \hat{i} - \frac{4}{3} \hat{j} - \frac{1}{3} \hat{k})

hence image of given point is

(- \frac{7}{3}, - \frac{4}{3}, - \frac{1}{3}) .

Commented by 1kanika# last updated on 09/Jul/17

Ans . of the ques . is (- 3, - 2, 0)

Commented by 1kanika# last updated on 09/Jul/17

Ans . of the ques . is (- 3, - 2, 0)

Commented by 1kanika# last updated on 09/Jul/17

Ans . of the ques . is (- 3, - 2, 0)

Commented by mrW1 last updated on 07/Jul/17

nice work!

Commented by mrW1 last updated on 09/Jul/17

distance from (1, 2, 3) to plane x + y + z = 1 is 5 / \sqrt{3}

distance from (- 7 / 3, - 4 / 3, - 1 / 3) to plane x + y + z = 1 is 5 / \sqrt{3}

distance from (- 3, - 2, 0) to plane x + y + z = 1 is 6 / \sqrt{3}

\Rightarrow ans . (- 7 / 3, - 4 / 3, - 1 / 3) is right!

\Rightarrow ans . (- 3, - 2, 0) is wrong!

Commented by 1kanika# last updated on 09/Jul/17

we find reflcetion of a point not

image of that point .