Question

If $\overline{a},\overline{b}$ and $\overline{c}$ are unit coplanar vector than the scalar triple product $[2\overline{a}-\overline{b}2\overline{b}-\overline{c}2\overline{c}-\overline{a}]$ is equal to

• A. $0$
• B. $1$
• C. $-\sqrt{3}$
• D. $\sqrt{3}$

Answer 1

Answer: (A)

$0$

$\hat{a},\hat{b}\hat{c}$ are coplaner vectors

So, $\hat{a}(\hat{b}\times \hat{c})=\hat{a}\hat{b}.(\hat{c}\times \hat{a})=0$

$\hat{c}(\hat{a}\times \hat{b})=0$

Now,

$(2\hat{a}-\hat{b})\left[\right(2\hat{b}-\hat{c})\times (2\hat{c}-\hat{a}\left)\right]=(2\hat{a}-\hat{b})[2\hat{b}\times (2\hat{c}-\hat{a})-\hat{c}\times (2\hat{c}-\hat{a}\left)\right]=(2\hat{a}-\hat{b})[4\hat{b}\times \hat{c}-2\hat{b}\times \hat{a}-\hat{c}\times 2\hat{c}+\hat{c}\times \hat{a}]=(2\hat{a}-\hat{b})[4\hat{b}\times \hat{c}+2\hat{a}\times \hat{b}+\hat{c}\times \hat{a}]=\hat{a}(\hat{b}\times \hat{c})-\hat{b}(4\hat{b}\times \hat{c})+2\hat{a}(\hat{a}\times \hat{b})-2\hat{b}(\hat{a}\times \hat{b})+2\hat{a}(\hat{c}\times \hat{a})-\hat{b}(\hat{c}\times \hat{a})=8\hat{a}(\hat{b}\times \hat{c})-0+0-0+0-\hat{b}(\hat{c}\times \hat{a})=8\hat{a}(\hat{b}\times \hat{c})-\hat{a}(\hat{b}\times \hat{c})=7\hat{a}(\hat{b}\times \hat{c})=0$