Question

if a vector, b vector ,c vector, d vector are coplanar vectors then show that ( a vector cross b vector)×(c vector ×d vector)=0​

Answer 1

Given :

Vector a , vector b , vector c and vector d are coplanar vectors .

To Prove  :

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

Solution :

Since , we know that cross product of two vectors is given as :

$\vec a \times \vec b = absin\theta$  

Here ,$\theta$ is the angle between $\vec a$ and $\vec b$ .

And here the resultant vector after cross multiplication is always perpendicular to both  the vectors a and b .

Also we have :

$\vec c\times \vec d = cd sin\theta'$  

And the resultant of cross product of vectors c and d is also perpendicular to the plane containing them .

Now since the resultant of $\vec a \times \vec b$ and $\vec c \times \vec d$ are parallel to each other , so the angle between them is 0.

So , $( \vec a \times \vec b ) \times ( \vec c \times \vec d ) = |\vec a \times \vec b||\vec c \times \vec d|sin0^0$

                                 = 0

Hence , the cross products  of $\vec a \times \vec b$ and $\vec c \times \vec d$ is 0 .