(Distance reduction to two dimensions)

MATHEMATICS · Multi-part · CBSE · Mathematics · Introduction to Three-Dimensional Geometry · Introduction to Three-Dimensional Geometry

Problem

(Distance reduction to two dimensions) (i) State the general formula for the distance between two points

P(x_1,y_1,z_1)

and

Q(x_2,y_2,z_2)

in three–dimensional space. (ii) Prove that if

z_1 = z_2

, the formula obtained in part (i) reduces to the familiar two–dimensional distance between the projections

P(x_1,y_1)

and

Q(x_2,y_2)

in the

xy

–plane. (iii) Using the result of part (ii), find the distance between the points

P(2,-3,5)

and

Q(7,4,5)

and verify that the same value is obtained when the three–dimensional formula of part (i) is applied directly.