Question

Find the distance between the following pairs of points:
(i) $(2,3,5)$ and $(4,3,1)$
(ii)$(-3,7,2)$and $\left(\right(2,4,-1)$
(iii) $(-1,3,-4)$ and $(1,-3,4)$
(iv) $(2,-1,3)$ and $(-2,1,3)$

Answer 1

The distance between points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$ is given by
$PQ=$ $\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2+{(z_2-z_1)}^2}$
(i) Distance between points $(2,3,5)$ and $(4,3,1)$
$=$ $\sqrt{{(4-2)}^2+{(3-3)}^2+{(1-5)}^2}$
$=$ $\sqrt{{\left(2\right)}^2+{\left(0\right)}^2+{(-4)}^2}$
$=$ $\sqrt{4+16}$
$=$ $\sqrt{20}$
$=$ $2\sqrt{5}$
(ii) Distance between points $(-3,7,2)$ and $(2,4,-1)$
$=$ $\sqrt{{(2+3)}^2+{(4-7)}^2+{(-1-2)}^2}$
$=$ $\sqrt{{\left(5\right)}^2+{(-3)}^2+{(-3)}^2}$
$=$ $\sqrt{25+9+9}$
$=$ $\sqrt{43}$
(iii) Distance between points $(-1,3,-4)$ and $(1,-3,4)$
$=$ $\sqrt{{(1+1)}^2+{(-3-3)}^2+{(4+4)}^2}$
$=$ $\sqrt{{\left(2\right)}^2+{(-6)}^2+{\left(8\right)}^2}$
$=$ $\sqrt{4+36+64}=\sqrt{104}$
$=2\sqrt{26}$
(iv) Distance between points $(2,-1,3)$ and $(-2,1,3)$
$=$ $\sqrt{{(-2-2)}^2+{(1+1)}^2+{(3-3)}^2}$
$=$ $\sqrt{{(-4)}^2+{\left(2\right)}^2+{\left(0\right)}^2}$
$=$ $\sqrt{16+4}$
$=$ $\sqrt{20}$
$=$ $2\sqrt{5}$