Question

If the vectors $3\widehat i + \widehat j - 5\widehat k$and $a\widehat i + b\widehat j - 15\widehat k$ are collinear, then,
1. $a = 3,b = 1$
2. $a = 9,b = 1$
3. $a = 3,b = 3$
4. $a = 9,b = 3$

Answer 1

Hint: In this question we are given that, two vectors $3\widehat i + \widehat j - 5\widehat k$ and $a\widehat i + b\widehat j - 15\widehat k$ are collinear and we have to find the value of $a,b$. Now to calculate the values take the first vector to be equal to $p$ times the second vector where $p$ is scalar. Now compare both sides and you’ll get the value of $p$. Put the required values in another equation and solve.

Formula Used:
If two vectors $x\widehat i + y\widehat j + z\widehat k$ and $p\widehat i + q\widehat j + r\widehat k$ are collinear
Then, $x\widehat i + y\widehat j + z\widehat k = k\left( {p\widehat i + q\widehat j + r\widehat k} \right)$

Complete step by step Solution:
Given that,
Vectors $3\widehat i + \widehat j - 5\widehat k$and $a\widehat i + b\widehat j - 15\widehat k$ are collinear
It implies that,
$3\widehat i + \widehat j - 5\widehat k = p\left( {a\widehat i + b\widehat j - 15\widehat k} \right)$here, $p$is scalar
$3\widehat i + \widehat j - 5\widehat k = ap\widehat i + bp\widehat j - 15p\widehat k - - - - - \left( 1 \right)$
Compare both the sides of equation (1)
$ \Rightarrow ap = 3 - - - - \left( 2 \right)$
$bp = 1 - - - - \left( 3 \right)$
$ - 5 = - 15p$
Therefore, $p = \dfrac{1}{3}$
Put the value of $ - 5 = - 15k$ in equation (2) and (3)
$ \Rightarrow a = 9, b = 3$

Hence, the correct option is 4.

Note: The key concept involved in solving this problem is a good knowledge of collinear vectors. Students must know that two vectors are collinear if and only if they are either parallel to one another in the same direction may be the opposite direction or are along the same line. Collinear vectors are regarded as an important concept in vector algebra. Collinear vectors are defined as two or more given vectors that lie along the same given line. Two parallel vectors can be considered collinear vectors because they point in the same or opposite direction.