derive the formula for distance between the point (x1,y1,z1) and (x2,y2,z2) Hence find the distance betweeen the point p(1,-3,4) - Maths - Introduction to Three Dimensional Geometry

Question

derive the formula for distance between the point
(x1,y1,z1) and
(x2,y2,z2) Hence find the distance
betweeen the point p(1,-3,4) and (-4,1,2)

Manbar Singh · Accepted Answer

(1)
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style="width: 360px; height: 395px;">

Let O be the origin
Let Px1,y1,z1 and Qx2,y2,z2 be the given points
From P and Q, draw perpendiculars PM and QN respectively on xy-plane
Then coordinates of M and N are Mx1,y1,0 and Nx2,y2,0
The 2-dimensional coordinates of M and N referred to OX nd OY are : Mx1,y1 and Nx2,y2
MN2 = x2-x12 + y2-y12From P, draw PR⊥QN
Now, PR ∥ MN and PR = MN
In ∆PRQ, PQ2 = PR2 + QR2   Pythagoras theoremPQ2 = MN2 + QN-RN2PQ2 =MN2 +  QN-PM2PQ2 = x2-x12 + y2-y12 + z2-z12   as, QN = z2; PM = z1PQ =  x2-x12 + y2-y12 + z2-z12

(2)

The given points are P1,-3,4 and Q-4,1,2Now, PQ = -4-12+1+32 + 2-42⇒PQ = 25 + 16 + 4 = 45 = 35 units

derive the formula for distance between the point (x1,y1,z1) and (x2,y2,z2). Hence find the distance betweeen the point p(1,-3,4) and (-4,1,2)

derive the formula for distance between the point (x_1,y_1,z₁) and (x_2,y_2,z₂). Hence find the distance betweeen the point p(1,-3,4) and (-4,1,2)