Question

If projections of a line segment on coordinate axes be \[2,\; - 1,\;2\] respectively, then, its length will be
A. $\dfrac{1}{2}$
B. $2$
C. $3$
D. $4$

Answer 1

Hint: In order to solve this type of question, first we will assume the line segment in vector form. Next, we will take the dot product of the line segment and the direction vector. Then, we will substitute the given values. Now, we will find the length of the line segment by finding the magnitude of the vector formed.

Formula used:
$\left| {\overrightarrow a } \right| = \sqrt {{a_1}^2 + {b_1}^2 + {c_1}^2} $

Complete step by step solution:
We are given that,
Projections of a line segment on coordinate axes are \[2,\; - 1,\;2\] ………………..equation $\left( 1 \right)$
Let us assume that the line segment be represented in vector form as
$\overrightarrow a = {a_1}\widehat i + {b_1}\widehat j + {c_1}\widehat k$ ………………..equation $\left( 2 \right)$
Vectors along coordinate axes are $\widehat i,\widehat j,\widehat k$ ………………..equation $\left( 3 \right)$
Taking the dot product of the line segment and the direction vector and equating it with equation $\left( 1 \right)$,
${a_1}\widehat i \times \widehat i = 2$
$ \Rightarrow {a_1} = 2$
Solving it for y-coordinate,
${b_1}\widehat j \times \widehat j = - 1$
$ \Rightarrow {b_1} = - 1$
Solving it for z-coordinate,
${c_1}\widehat k \times \widehat k = 2$
$ \Rightarrow {c_1} = 2$
Substituting these values in equation $\left( 2 \right)$,
$\overrightarrow a = 2\widehat i + \left( { - 1} \right)\widehat j + 2\widehat k$
Finding the length of $\overrightarrow a $,
$\left| {\overrightarrow a } \right| = \sqrt {{a_1}^2 + {b_1}^2 + {c_1}^2} $
Substituting the values,
$\left| {\overrightarrow a } \right| = \sqrt {{2^2} + {{\left( { - 1} \right)}^2} + {2^2}} $
Solving it,
$\left| {\overrightarrow a } \right| = \sqrt 9 $
$\left| {\overrightarrow a } \right| = 3$
Thus, the length of the line segment is 3 units.
$\therefore $ The correct option is (C).

Note: The key concept to solve this type of question is to start with assuming the line segment in vector form. Use $\widehat i \times \widehat i = 1,$ $\widehat j \times \widehat j = 1$ and $\widehat k \times \widehat k = 1$ while taking the dot product and solving it. Also, be sure of the calculations to avoid any unnecessary mistakes in order to get the correct answer.