Question

Let a = 2i +3j - k and b =(i-2j +3k) then the value of l for which the vector c=li+j+(2l-1)k is parallel to plane containing a and b is
(a) 1
(c) -1
(b) 0
(d) 2​

Answer 1

Given :

$\vec a$ = 2i+ 3j - k

$\vec b$ = i - 2j + 3k

$\vec c$ =li + j + (2l-1) k

Vector c is parallel to plane containing vectors a and b.

To Find :

Value of l = ?

Solution :

We know that vector $\vec a \times \vec b$ will be perpendicular to the plane containing vectors a and b .

If $\vec c$ is parallel to the plane containing $\vec a$ and $\vec b$ then we have ;

$\vec c. ( \vec a \times \vec b)= 0$

This is because $\vec c$ would make an angle of 90°  with the vector $\vec a \times \vec b$ .

Since ,$\vec a \times \vec b = \left|\begin{array}{ccc}i&j&k\\2&3&-1\\1&-2&3\end{array}\right|$

                   = $( 7i -7j - 7k )$

Now , $\vec c . (\vec a \times \vec b) = [li + j + (2l -1)k].(7i -7j -7k )$

                         =$7l - 7 - 7(2l -1 )$

Also we have , $\vec c . (\vec a \times \vec b) = 0$

So, 7l - 7 - 7(2l-1) = 0

Or, 7l - 7 - 14l + 7 = 0

Or, 7l - 14l = 0

Or , l = 0

So , the value of l is 0 .