Name: Equation of a Line in 3D (Symmetric Form)
Uploaded: 2019-07-19
Duration: 11 min 8 s
Description: We have seen the equation of a line in 2D. Here, in this video, we will see the equation of a line in 3D and also see why vectors play an important role in 3...

Question

Find the coordinates of the point which is equidistant from the points

A (a, 0, 0), B (0, b, 0), C (0, 0, c)

and

D (0, 0, 0)

.

Accepted Answer

Given points $A (a, 0, 0), B (0, b, 0), C (0, 0, c)$ and $D (0, 0, 0)$ .

Let point $P (x, y, z)$ which is equidistant from given all points.

So, $P A = P B = P C = P D$

Using distance formula, Distance $= \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z_{1})}^{2}}$

Now, $P A = P D$ $\Rightarrow P A^{2} = P D^{2}$

$\Rightarrow {(x - a)}^{2} + {(y - 0)}^{2} + {(z - 0)}^{2} = {(x - 0)}^{2} + {(y - 0)}^{2} + {(z - 0)}^{2}$

$\Rightarrow x^{2} + a^{2} - 2 a x + y^{2} + z^{2} = x^{2} + y^{2} + z^{2} \Rightarrow a^{2} - 2 a x = 0 \Rightarrow x = \frac{a}{2}$

Now, $P B^{2} = P D^{2}$

$\Rightarrow {(x - 0)}^{2} + {(y - b)}^{2} + {(z - 0)}^{2} = {(x - 0)}^{2} + {(y - 0)}^{2} + {(z - 0)}^{2} \Rightarrow x^{2} + y^{2} + b^{2} - 2 b y + z^{2} = x^{2} + y^{2} + z^{2} \Rightarrow b^{2} - 2 b y = 0 \Rightarrow y = \frac{b}{2}$

Now, $P C^{2} = P D^{2}$

$\Rightarrow {(x - 0)}^{2} + {(y - 0)}^{2} + {(z - c)}^{2} = {(x - 0)}^{2} + {(y - 0)}^{2} + {(z - 0)}^{2} \Rightarrow x^{2} + y^{2} + z^{2} + c^{2} - 2 c z = x^{2} + y^{2} + z^{2} \Rightarrow 2 c z = c^{2} \Rightarrow z = \frac{c}{2}$

Hence point is $P (x, y, z) = P (\frac{a}{2}, \frac{b}{2}, \frac{c}{2})$ .

Find the coordinates of the point which is equidistant from the points $A (a, 0, 0), B (0, b, 0), C (0, 0, c)$ and $D (0, 0, 0)$ .

Important Questions on Introduction to Three Dimensional Geometry

Learn and Prepare for any exam you want !

Find the coordinates of the point which is equidistant from the points Aa,0,0,B0,b,0,C0,0,c and D0,0,0.

Important Questions on Introduction to Three Dimensional Geometry

Learn and Prepare for any exam you want !

Find the coordinates of the point which is equidistant from the points $A (a, 0, 0), B (0, b, 0), C (0, 0, c)$ and $D (0, 0, 0)$ .