Question

Q.1. If the vectors i -2j+ k, ai – 5j + 3k & 5i - 9j + 4kare coplanar, then the value of a is. a) 3. b)2. c) -3. d)-2 ((≧∇≦))​

Answer 1

Step-by-step explanation:

Goven vectors be

$\tt{ \vec{a} = \hat{i} - 2 \hat{j} + \hat{k}} \\ \tt{ \: \: \: \: \: \vec{b} = a\hat{i} - 5 \hat{j} + 3 \hat{k}} \\ \tt{ \: \: \: \: \: \vec{c} = 5 \hat{i} -9 \hat{j} + 4\hat{k}}$

Since they are coplaner, so,

$\left[ \begin{array}{r} \vec{a} & \vec{b} &\vec{c}\end{array}\right] = 0$

$\implies\left| \begin{array}{ccc} 1 & - 2 &1 \\a & - 5 &3 \\ 5 & -9 &4\end{array}\right| = 0$

$\implies1\left| \begin{array}{cc} - 5 &3 \\ -9 &4\end{array}\right| - ( - 2)\left| \begin{array}{cc} a &3 \\ 5 &4\end{array}\right| + 1\left| \begin{array}{cc} a & - 5\\ 5& - 9 \end{array}\right|= 0$

$\implies1( - 20 + 27) + 2(4a - 15)+ 1( - 9a + 25)= 0$

$\implies7 + 8a - 30 - 9a + 25= 0$

$\implies7 - a - 5= 0$

$\implies2 - a = 0$

$\implies a =2$