Question

What are the Direction Ratios (DR’s) of the vector which is parallel to \[\left( {2, - 1,1} \right)\] and \[\left( {3,4, - 1} \right)\] respectively?
(a) \[\left( { - 2,5,1} \right)\]
(b) \[\left( {1,5, - 2} \right)\]
(c) Cannot be determined
(d) None of these

Answer 1

Hint: The given problem revolves around the concepts of vector geometry. As a result, considering the position vectors of the given values, comparing it with the definition of directional ratios, and subtracting other values from the first one, the desired solution is obtained.

Complete step-by-step answer:
Since, let us assume some of the data for calculating Direction Ratios that is (DR’s) from the given values,
Let ‘\[\bar a\]’ and ‘\[\bar b\]’ be the position vectors of the given vertices ‘\[\left( {2, - 1,1} \right)\]’ and ‘\[\left( {3,4, - 1} \right)\]’ of the points ‘\[A\]’ and ‘\[B\]’ respectively.
That is,
\[A \equiv \left( {2, - 1,1} \right)\]
And,
\[B \equiv \left( {3,4, - 1} \right)\]
Hence, we get
\[\vec a = 2i - j + k\] … (i)
And,
\[\vec b = 3i + 4j - k\] … (ii)
But, we have given that the vectors are parallel as a result, also, known as “Collinear vector” in terms of vector geometry defines the (three) magnitudes in proportional say, a, b, c along with Direction Cosines (that is along the axes) say, \[\alpha \], \[\beta \], and \[\delta \] respectively, known as “Direction Ratios (DR’s)”.
Hence, from the definition
Subtracting the equation (ii) from (i), we get
\[\overrightarrow {AB} = \bar b - \bar a\]
Substituting the values in the equation, we get
\[\overrightarrow {AB} = \left( {3i + 4j - k} \right) - \left( {2i - j + k} \right)\]
Solving the equation mathematically, we get
\[\overrightarrow {AB} = 3i + 4j - k - 2i + j - k\]
\[\overrightarrow {AB} = i + 5j - 2k\] … (iii)
Where, the vertices of ‘\[AB\]’ exists the given condition i.e.
\[AB \equiv \left( {1,5, - 2} \right)\]
\[\therefore \Rightarrow \]The option (b) is absolutely correct.
So, the correct answer is “Option b”.

Note: One must know the vector geometry concept or even the basic definition which clarifies the idea to solve the question asked to find the position vectors which is based on magnitude and direction. As a result, knowing the definition of directional ratio will help easily to get the desired value, so as to be sure of our final answer.