Question

: The distance between the points (4, 3, 7) and (1, -1, -5) is?
(a) 7
(b) 12
(c) 13
(d) 25

Answer 1

Hint: Assume (4, 3, 7) as $(x_1,y_1,z_1)$ and (1, -1, 5) as $(x_2,y_2,z_2)$. Use distance formula in 3-dimensional geometry given by: $d=\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2+{(z_1-z_2)}^2}$, where ‘d’ is the distance between the two points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$.

Complete step-by-step answer:
We know that a point lying in any plane is represented by the coordinates $(x,y,z)$. Now, we have been provided with two points (4, 3, 7) and (1, -1, -5) and we have to find the distance between these two.
Let us assume these points as $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ respectively. Therefore,
$(4,3,7)=(x_1,y_1,z_1)$ and $(1,-1,-5)=(x_2,y_2,z_2)$.
Let us assume that the distance between these two points is $d$.
By distance formula, we know that, distance between two points is, $d=\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2+{(z_1-z_2)}^2}$. Therefore,
$\begin{array}{rl}& d=\sqrt{{(4-1)}^2+{(3-(-1))}^2+{(7-(-5))}^2}\\ & =\sqrt{3^2+4^2+{12}^2}\\ & =\sqrt{9+16+144}\\ & =\sqrt{169}\\ & =13\end{array}$
Therefore, the distance between these two points is 13 units.
Hence, option (c) is the correct answer.

Note: One may note that in 2-dimension geometry, we have only two coordinates of a particular point, that is, (x, y) which lies in the x-y plane. The distance between any two points in 2-D space is given by: $d=\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2}$. Similarly, in 3-D space we have another coordinate in addition to x and y, that is z. This z-coordinate represents that the required point is above or below the x-y plane. So, we use the distance formula, $d=\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2+{(z_1-z_2)}^2}$ for the calculation of distance between two points, like we did in the above question.