If bar{a}=3 hat{imath}+hat{jmath}-hat{k}, bar | vedclass

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(A) To determine if the vectors are coplanar,we calculate the scalar triple product $[\bar{a} \ \bar{b} \ \bar{c}]$.If the scalar triple product is $0

Vedclass · Accepted Answer

$\bar{a}, \bar{b}$ and $\bar{c}$ are non-coplanar.

Vedclass · Answer

(A) To determine if the vectors are coplanar,we calculate the scalar triple product $[\bar{a} \ \bar{b} \ \bar{c}]$.If the scalar triple product is $0$,the vectors are coplanar. If it is non-zero,they are non-coplanar.Given $\bar{a} = 3\hat{\imath} + \hat{\jmath} - \hat{k}$,$\bar{b} = 2\hat{\imath} - \hat{\jmath} + 7\hat{k}$,and $\bar{c} = 7\hat{\imath} - \hat{\jmath} + 23\hat{k}$.$[\bar{a} \ \bar{b} \ \bar{c}] = \begin{vmatrix} 3 & 1 & -1 \ 2 & -1 & 7 \ 7 & -1 & 23 \end{vmatrix}$$= 3((-1)(23) - (7)(-1)) - 1((2)(23) - (7)(7)) - 1((2)(-1) - (7)(-1))$$= 3(-23 + 7) - 1(46 - 49) - 1(-2 + 7)$$= 3(-16) - 1(-3) - 1(5)$$= -48 + 3 - 5 = -50$.Since the scalar triple product is $-50 
eq 0$,the vectors $\bar{a}, \bar{b},$ and $\bar{c}$ are non-coplanar. Thus,statement $A$ is true.

If $\bar{a}=3 \hat{\imath}+\hat{\jmath}-\hat{k}, \bar{b}=2 \hat{\imath}-\hat{\jmath}+7 \hat{k}$ and $\bar{c}=7 \hat{\imath}-\hat{\jmath}+23 \hat{k}$ are three vectors,then which of the following statements is true?

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