If the vectors 2hat{i}-hat{j}+3hat{k},hat{i}+ | vedclass

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(C) Three vectors $\vec{a}$,$\vec{b}$,and $\vec{c}$ are coplanar if and only if their scalar triple product is zero,i.e.,$[\vec{a} \vec{b} \vec{c}] =

Vedclass · Accepted Answer

$43$

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(C) Three vectors $\vec{a}$,$\vec{b}$,and $\vec{c}$ are coplanar if and only if their scalar triple product is zero,i.e.,$[\vec{a} \vec{b} \vec{c}] = 0$.This is equivalent to the determinant of the matrix formed by their components being zero:$\begin{vmatrix} 2 & -1 & 3 \ 1 & 4 & 1 \ 4 & p & 1 \end{vmatrix} = 0$Expanding the determinant along the first row:$2(4(1) - 1(p)) - (-1)(1(1) - 1(4)) + 3(1(p) - 4(4)) = 0$$2(4 - p) + 1(1 - 4) + 3(p - 16) = 0$$8 - 2p - 3 + 3p - 48 = 0$$p - 43 = 0$$p = 43$Thus,the correct option is $C$.

If the vectors $2\hat{i}-\hat{j}+3\hat{k}$,$\hat{i}+4\hat{j}+\hat{k}$,and $4\hat{i}+p\hat{j}+\hat{k}$ are coplanar,then $p=$

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Consider the vectors $\vec{a}=2 \hat{i}+3 \hat{j}-6 \hat{k}$, $\vec{b}=6 \hat{i}-2 \hat{j}+3 \hat{k}$ and $\vec{c}=3 \hat{i}-6 \hat{j}-2 \hat{k}$.
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