Three vectors $\vec A,\vec B,\vec C$ Compare the value of these 4 questions.(use the parallelepiped)
a) $(\vec A\times \vec C)\cdot\vec B$
b) $\vec A\cdot(\vec C\times\vec B)$
c) $(\vec A\times\vec B)\cdot\vec C$
d) $\vec B\cdot(\hat A \times \vec A)$
Where $\hat A=\frac{1}{A}\vec A$. I solve this question but something makes me confusing.
First answer $a=b<d<c$
Second answer $a=b≠c,d=0$
But I think second answer is best answer because, Three vectors about sides of parallelepiped can change each other. So,a),b) is volume of parallelepiped but it makes ±V and c) is decoding order V Which is the best answer first or second ? please solve this question.
To get some intuition it's always good to draw a picture, especially for questions like this.
This drawing show the parallelepiped formed by $\vec A,\vec B,\vec C$. The volume is $(\vec A\times\vec B)\cdot \vec C=V$. The volume is positive in my example because these three vectors follow the right hand rule: if I take the cross product of two (for example $\vec A$ and $\vec B$) then the result will point in the general direction of the third vector (vector $\vec C$ ). From this picture we can conclude that if we switch the vectors in such a way that the right hand rule is still obeyed then the volume stays the same. In other words $$(\vec A\times\vec B)\cdot \vec C=(\vec B\times\vec C)\cdot \vec A=(\vec C\times\vec A)\cdot \vec B$$
The dot product obeys $\vec u\cdot \vec v=\vec v\cdot \vec u$ so we have $$(\vec A\times\vec B)\cdot \vec C=\vec C\cdot(\vec A\times\vec B)$$
The cross product obeys $$\vec A\times\vec B=-\vec B\times\vec A$$
Using these relations you should be able to relate $a,b,c$ to eachother and they only differ by a minus sign.
Question d) is different. To see how this works look at the picture and imagine what happens if I move $\vec B$ until it points in the direction of $\vec A$. What happens to the volume?