Question

If the three vectors are coplanar, then find the value of ′x′ is
\[\overrightarrow A = \widehat i - 2\widehat j + 3\widehat k\]
\[\overrightarrow B = x\widehat j + 3\widehat k\]
\[\overrightarrow C = 7\widehat i + 3\widehat j - 11\widehat k\]
A. \[\dfrac{{36}}{{21}}\]
B. \[ - \dfrac{{51}}{{32}}\]
C. \[\dfrac{{51}}{{32}}\]
D. \[ - \dfrac{{36}}{{21}}\]

Answer 1

Hint:Before we start addressing the problem, we need to know about what data has been provided and what we need to solve. Here the three vectors are given which is coplanar which means all the vectors lie on the same plane and we need to find the value of x. This we can find using the formula when the three vectors are coplanar then their determinant of coefficient of all the three vectors is equal to zero.

Formula Used:
The formula to find when the three vectors are coplanar, then
\[\left| {ABC} \right| = 0\]……… (1)
Where, A, B and C are the vectors.

Complete step by step solution:
If \[\overrightarrow A \], \[\overrightarrow B \]and \[\overrightarrow C \] are coplanar vectors then, the determinants of the coefficient of all the vectors are equal to zero. Therefore, from equation (1) we have,
\[\left| {ABC} \right| = 0\]
\[ \Rightarrow \left| {\begin{array}{*{20}{c}}1&{ - 2}&3\\0&x&3\\7&3&{ - 11}\end{array}} \right| = 0\]
\[ \Rightarrow 1\left( { - 11x - 9} \right) + 2\left( {0 - 21} \right) + 3\left( {0 - 7x} \right) = 0\]
\[ \Rightarrow - 32x = 51\]
\[ \therefore x = \dfrac{{ - 51}}{{32}}\]
Therefore, the value of x is \[\dfrac{{ - 51}}{{32}}\].

Hence, option B is the correct answer.

Additional information: The conditions of coplanar vectors are;
1. Three vectors in 3D space can be said to be coplanar when their scalar triple product is 0.
2. If the three vectors are coplanar, they should be linearly dependent.
There are different types of vectors : zero vector, unit vector, position vector, co-initial vector, coplanar vector, collinear vector, equal vector etc.

Note: Don’t get confused with the collinear vector and coplanar vectors. The collinear vectors are those which lie parallel to the same line irrespective of their magnitudes. Coplanar vectors are one in which all the vectors lie on the same plane.