If A(3,1,-1),B((5)/(3),(7)/(3),(1)/(3)),C(2,2,1) and D((10)/(3),(2)/(3),(-1)/(3)) are the …

Question

If $A(3,1,-1),B(\frac{5}{3},\frac{7}{3},\frac{1}{3}),C(2,2,1)$ and $D(\frac{10}{3},\frac{2}{3},\frac{-1}{3})$ are the vertices of a quadrilateral $ABCD$, then its area is

Accepted Answer

$A(3,1,-1),B(\frac{5}{3},\frac{7}{3},\frac{1}{3}),C(2,2,1),D(\frac{10}{3},\frac{2}{3},\frac{-1}{3})$ are vertices of a quadrilateral

$\begin{gathered} \overset{	o \leftarrow }{AC}=(2\hat{i}+2\hat{j}+\hat{k})-(3\hat{i}+\hat{j}-\hat{k}) \ =-\hat{i}+\hat{j}+2\hat{k} \ \overset{	o \leftarrow }{BD}=(\frac{10}{3}\hat{i}+\frac{2}{3}\hat{j}-\frac{1}{3}\hat{k})-(\frac{5}{3}\hat{i}+\frac{7}{3}\hat{j}+\frac{1}{3}\hat{k}) \ \overset{	o \leftarrow }{BD}=\frac{5}{3}\hat{i}-\frac{5}{3}\hat{j}-\frac{2}{3}\hat{k} \ =\frac{1}{2}|\overset{	o \leftarrow }{AC}	imes \overset{	o \leftarrow }{BD}| \ =\frac{1}{2}|\begin{array}{ccc}\hat{i} \quad  \hat{j} \quad  \hat{k} \ -1 \quad  1 \quad  2 \ \frac{5}{3} \quad  \frac{-5}{3} \quad  \frac{-2}{3}\end{array}| \ =\frac{1}{2}\sqrt{{(\frac{8}{3})}^{2}+{(\frac{8}{3})}^{2}}[\because \overset{	o \leftarrow }{AC}	imes \overset{	o \leftarrow }{BD}=\frac{8}{3}\hat{i}+\frac{8}{3}\hat{j}] \ =\frac{1}{2}	imes \frac{8}{3}	imes \sqrt{2}=\frac{4\sqrt{2}}{3} \end{gathered}$

Solutions · 1