Question

$\left(i\right)$ Find the distance between the points $(2,3,5)\text{and}(4,3,1)$

$\left(ii\right)$ Find the distance between the points $(-3,7,2)\text{and}(2,4,-1)$

$\left(iii\right)$ Find the distance between the points $(-1,3,-4)\text{and}(1,-3,4)$

$\left(iv\right)$ Find the distance between the points $(2,-1,3)\text{and}(-2,1,3)$

Answer 1

$\left(i\right)$ Given : Points $(2,3,5)\text{and}(4,3,1)$
Let $P(2,3,5)\text{and}Q(4,3,1)$ be the points.

Distance $PQ$
$PQ=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$

Here,
$x_1=2,y_1=3,z_1=5$
$x_2=4,y_2=3,z_2=1$

$PQ=\sqrt{(4-2{)}^2+(3-3{)}^2+(1-5{)}^2}$
$=\sqrt{4+0+16}$
$=\sqrt{20}$
$=2\sqrt{5}$

Thus, the required distance is $2\sqrt{5}\text{units}$

$\left(ii\right)$ Given : Points $(-3,7,2)\text{and}(2,4,-1)$
Let $P(-3,7,2)\text{and}Q(2,4,-1)$ be the points.

Distance $PQ$
$PQ=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$

Here,
$x_1=-3,y_1=7,z_1=2$
$x_2=2,y_2=4,z_2=-1$

$PQ=\sqrt{(2-(-3){)}^2+(4-7{)}^2+(-1-2{)}^2}$
$=\sqrt{25+9+9}$
$=\sqrt{43}$

Thus, the required distance is $\sqrt{43}\text{units}$

$\left(iii\right)$ Given : Points $(-1,3,-4)\text{and}(1,-3,4)$
Let $P(-1,3,-4)\text{and}Q(1,-3,4)$ be the points.

Distance $PQ$
$PQ=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$

Here,
$x_1=-1,y_1=3,z_1=-4$
$x_2=1,y_2=-3,z_2=4$

$PQ=\sqrt{(-1-1{)}^2+(-3-3{)}^2+(4+4{)}^2}$
$=\sqrt{4+36+64}$
$=\sqrt{104}$
$=2\sqrt{26}$

Thus, the required distance is $2\sqrt{26}\text{units}$

$\left(iv\right)$ Given : Points $(2,-1,3)\text{and}(-2,1,3)$
Let $P(2,-1,3)\text{and}Q(-2,1,3)$ be the points.

Distance $PQ$
$PQ=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$

Here,
$x_1=2,y_1=-1,z_1=3$
$x_2=-2,y_2=1,z_2=3$

$PQ=\sqrt{(-2-2{)}^2+(1-(-1){)}^2+(3-3{)}^2}$
$=\sqrt{16+4+0}$
$=\sqrt{20}$
$=2\sqrt{5}$

Thus, the required distance is $2\sqrt{5}\text{units}$