Question

Prove that the vectors A=6i+9j-12k and B=2i+3j-4k are parallel.

Answer 1

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Answer 2

Answer:

The given two vectors are parallel to each other.

Explanation:

Given two vectors:

$\vec A=6\hat i+9\hat j-12\hat k$  and  $\vec B=2\hat i+3\hat j-4\hat k$

It the two vectors are parallel to each other, then the angle between them is zero.

The cross product of two vectors is given by

$\vec A\times \vec B=|\vec A|| \vec B|sin\theta$

When the angle between the vectors is zero, $\theta = 0^o$

Then $sin\theta=sin0^o=0$

Therefore, $\vec A\times \vec B=0$

$\vec A\times \vec B=\left|\begin{array}{ccc}\hat i&\hat j&\hat k\\6&9&-12\\2&3&-4\end{array}\right|$

         $=(9\times (-4)-(-12)\times 3)\hat i-(6\times (-4)-(-12)\times 2)\hat j+(6\times 3-9\times 2)\hat k$

        $=(-36+36)\hat i-(-24+24)\hat j+(18-18)\hat k$

        $=0$

Since, $\vec A\times \vec B=0$

Therefore, the given two vectors are parallel to each other.