Question

Let \[\vec \alpha = \left( {\lambda - 2} \right)\vec a + \vec b\] and \[\vec \beta = \left( {4\lambda - 2} \right)\vec a + \vec b\] be two given vectors when vectors \[\vec a\] and \[\vec b\] are non-collinear. Find the value of \[\left| \lambda \right|\] for which vectors \[\vec \alpha \] and \[\vec \beta \] are collinear.
A. \[3\]
B. \[ - 4\]
C. \[6\]
D. \[9\]

Answer 1

Hint: In the given question, we need to find the value of \[\left| \lambda \right|\]. For this, we will use the property of collinear vectors such as two vectors are collinear if and only if relations of their coordinates are equal to get the desired result.

Formula used: The following formula used for solving the given question.
 If \[\vec u = a\hat i + b\hat j + c\hat k,\vec v = p\hat i + q\hat j + r\hat k\] are collinear then \[\dfrac{a}{p} = \dfrac{b}{q} = \dfrac{c}{r}\]

Complete step by step solution: We know that \[\vec \alpha = \left( {\lambda - 2} \right)\vec a + \vec b\] and \[\vec \beta = \left( {4\lambda - 2} \right)\vec a + \vec b\]
Here, vectors \[\vec a\] and \[\vec b\] are non-collinear.
We know that two vectors are collinear if relations of their coordinates are equal.
So, we get
\[\dfrac{{\left( {\lambda - 2} \right)}}{{\left( {4\lambda - 2} \right)}} = \dfrac{1}{3}\]
By simplifying, we get
\[3\left( {\lambda - 2} \right) = \left( {4\lambda - 2} \right)\]
\[3\lambda - 6 = 4\lambda - 2\]
By simplifying further, we get
\[\lambda = - 4\]
Hence, the value of \[\left| \lambda \right|\] for which vectors \[\vec \alpha \] and \[\vec \beta \] are collinear, is \[ - 4\].

Thus, Option (B) is correct.

Additional Information: The definition of a vector is an entity with both magnitude and direction. The movement of an object between two points is described by a vector. The directed line segment can be used to geometrically represent vector mathematics. The magnitude of a vector is the length of the directed line segment, and the vector's direction is indicated by the angle at which it is inclined. A vector's "Tail" (the point where it begins) and "Head" (the point where it ends and has an arrow) are its respective names.


Note: Many students make mistake in simplification part as well as writing the property of collinear vectors. This is the only way, through which we can solve the example in simplest way. Also, it is necessary to find the appropriate value of \[\lambda \] using the expression \[\dfrac{{\left( {\lambda - 2} \right)}}{{\left( {4\lambda - 2} \right)}} = \dfrac{1}{3}\].