Find the value of a for which the vectors \(2\hat i - 3\hat j + 4\hat k\;\ and\ \;a\hat i + 6\hat j - 8\hat k\) are collinear?

CONCEPT:

For two vectors \(\vec m \ and \ \vec n \) to be collinear,​ \(\vec m\; = \;λ \vec n\) where λ is a scalar.

CALCULATION:

Given: The vectors \(2\hat i - 3\hat j + 4\hat k\;\ and\ \;a\hat i + 6\hat j - 8\hat k\) are collinear.

As we know that, if two vectors \(\vec m \ and \ \vec n \) are collinear then \(\vec m\; = \;λ \vec n\) where λ is a scalar.

⇒ \(2\hat i - 3\hat j + 4\hat k\;\ = λ × (\;a\hat i + 6\hat j - 8\hat k)\)

⇒ \(2\hat i - 3\hat j + 4\hat k\;\ = λ a \hat i + 6λ \hat j - 8λ \hat k\)

⇒ λa = 2 and 6λ = - 3

⇒  λ = -1/2

So, by substituting λ = -1/2 in the equation λa = 2 we get

⇒ a = 2 × - 2 = - 4

Hence, option B is the correct answer.