Question

Write distance of a point from \[p\left( {a,b,c} \right)\] from x-axis.

Answer 1

Hint: Whenever it says distance of point it means perpendicular distance.
we will use 3d co-ordinate geometry to solve this question.
And for this we have a formula of finding distance between two points.
If points are given as ${p_1}({a_1},{b_1},{c_1})$ and ${p_2}({a_2},{b_2},{c_2})$ then distance between them will be ${p_2}{p_1} = \sqrt {(a_2^2 - a_1^2) + (b_2^2 - b_1^2) + (c_2^2 - c_1^2)} $

Complete step by step answer:
We have ${p_1}$ as $p(a,b,c)$ we need to find ${p_2}$ so x-axis ${p_2}$ will be $(a,0,0)$ because we are calculating perpendicular distance. So, its x-coordinate will be a and y, z coordinates will be O.
To learn this concept follow this diagram.

From above diagram it is clear the distance of point p from x axis is a because or x-axis, y and z coordinates are O.
Here we got our second point we can name it ${p_2}$. So ${p_1}(a,0,0)$ and ${p_2}(a,b,c)$
By using distance formula we will find out distance of x-axis from point $p(a,b,c)$.
We know that ${p_2}{p_1} = \sqrt {{{({a_2} - {a_1})}^2} + {{({b_2} - {b_1})}^2} + {{({c_2} - {c_1})}^2}} $
So, ${p_2}{p_1} = \sqrt {{{(a - a)}^2} + {{(0 - b)}^2} + {{(0 - c)}^2}} $
${p_2}{p_1} = \sqrt {{b^2} + {c^2}} $
So answer of above question is distance of a point $p(a,b,c)$ from x-axis = $\sqrt {{b^2} + {c^2}} $

Note:
Abscissa or the x-coordinate of a point is its distance from the y-axis and the ordinate or the y-coordinate is its distance from the x-axis.
Coordinates of a point on the x-axis are of the form \[\left( {x,0} \right)\] and that of the point on the y-axis is of the form \[\left( {0,{\text{ }}y} \right),\] and on z-axis it is in the form of \[\left( {0,0,z} \right)\]
We can directly use the following formulas
Distance from x-axis $ = \sqrt {{y^2} + {z^2}} $
Distance from y-axis $ = \sqrt {{x^2} + {z^2}} $
Distance from z-axis $ = \sqrt {{x^2} + {y^2}} $