Question

The vectors which is/are coplanar with vectors $i + j + 2k$ and $i + 2j + k$ , are perpendicular to the vector $i + j + k$ is/are:

Answer 1

Any vector can be expressed as a linear combination of the other two vectors If the vectors are coplanar, If three vectors are coplanar these are the conditions. 1) their scalar product is zero and If they are existing in zero space.2) If the vectors are in 3d and linearly independent 3) If more than two vectors are linearly independent then all the vectors are coplanar.

Complete step by step solution:
Coplanar vectors can be defined as vectors that are placed on the same three-dimensional plane. The vectors are parallel to the same plane. To find any two arbitrary vectors in a plane, which are coplanar, is easy. The Coplanarity of the two lines lies in a three-dimensional space that is represented in a vector form.
Let us consider the given vector quantities $a{\text{ }} = {\text{ }}i + j + 2k$ and $b = {\text{ }}i + 2j + k$
$c{\text{ }} = {\text{ }}i + j + k$
Let the vector on the plane of a and b is in the form of
$r{\text{ }} = {\text{ }}\lambda a{\text{ }} + {\text{ }}\mu b$
$\;\;\;\; = \lambda \left( {i + j + 2k} \right){\text{ }} + {\text{ }}\mu \left( {i + 2j + k} \right)$
$= {\text{ }}\left( {\lambda + \mu } \right)i + \left( {\lambda + 2\mu } \right)j + \left( {2\lambda + \mu } \right)k$
Also considering, $r.c{\text{ }} = {\text{ }}0$
Hence taking the values of $r.c{\text{ }}$from above, We can get the value as $\left( {\lambda + \mu } \right).1 + \left( {\lambda + 2\mu } \right).1 + \left( {2\lambda + \mu } \right){\text{ }}.{\text{ }}1{\text{ }} = {\text{ }}0$
$4\lambda {\text{ }} + {\text{ }}4\mu {\text{ }} = {\text{ }}0$
$\lambda {\text{ }} + {\text{ }}\mu {\text{ }} = {\text{ }}0$Hence we get
$\left[ {r{\text{ }}a{\text{ }}b} \right]{\text{ }} = {\text{ }}0$
Therefore, the two vectors $i - j$and $- j + k$ satisfy the coplanar conditions.
Hence. Both options 3)$i - j$ and 4) $- j + k$ are correct since they both satisfy the required conditions

Note: A linear combination of vectors ${v_1},{\text{ }} \ldots ,{\text{ }}{v_n}\;$with coefficients ${a_1},{\text{ }} \ldots ,{\text{ }}{a_n}\;$is a vector, such that ${a_1}{v_1}\; + {\text{ }} \ldots {\text{ }} + {\text{ }}{a_n}{v_n}$A linear combination ${a_1}{v_1}\; + {\text{ }} \ldots {\text{ }} + {\text{ }}{a_n}{v_n}\;$is called trivial if all the coefficients${\text{ }}{a_1},{\text{ }} \ldots ,{\text{ }}{a_n}$ are zero, and if at least one of the coefficients is not zero, then it is called non-trivial. Two lines can be called coplanar when they both lie on the same plane in a three-dimensional space.