Question

If $A\left( {1,2,3} \right),B\left( { - 1, - 1, - 1} \right)$ be the points, then the distance AB is
A. $\sqrt 5 $
B. $\sqrt {21} $
C. $\sqrt {29} $
D. None of these

Answer 1

Hint: As in this question we are given two coordinates of two points. Now to find the distance between these coordinates we can use the distance formula.

Formula Used:
Distance between two points$=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$

Complete step-by-step solution:
Given: AB is the line given with coordinates of $A\left( {1,2,3} \right)$and$B\left( { - 1, - 1, - 1} \right)$
To find the distance between AB we will use distance formula
Distance=$\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2} + {{\left( {{z_2} - {z_1}} \right)}^2}} $
 Inserting the value of coordinates A and B in above formula
Finding AB,
$AB = \sqrt {{{\left( { - 1 - 1} \right)}^2} + {{\left( { - 1 - 2} \right)}^2} + {{\left( { - 1 - 3} \right)}^2}} $
$AB = \sqrt {4 + 9 + 16} $
$ \Rightarrow AB = \sqrt {29} $
So, option C is correct.

Additional Information:
As its name implies, any distance formula provides the distance (the length of the line segment). For instance, the length of the line segment that joins two points represents the distance between them. We obtain the formula for distance between two points in a two-dimensional plane using the Pythagoras theorem, which can also be used to estimate the distance between two points on a three-dimensional plane.
The following are key ideas about the distance formula.
1. The distance determined using the distance formula is always positive.
2. The distance formula can be used to determine how far any point is from the origin.
3. The measured distance between the two places is the shortest linear distance.
4. For points located in any of the four quadrants, the distance calculation yields the same result.

Note: These types of questions are easy to solve if we understand the meaning of the question and use the correct formula. As there are various formulas for distance available but to use them we need to know what is given in the question.