Question

what is the area of the triangle with vertices at the points ( p=(1,2,3) ), ( q=(3,3,3) ), and ( r=(1,2,5) )?
(a) ( 2 sqrt{5} ).
(b) 5.
(c) ( sqrt{10} ).
(d) 10.
(e) ( sqrt{5} ).

Explanation:

Step1: Find vectors \(\overrightarrow{PQ}\) and \(\overrightarrow{PR}\)

Given \(P=(1,2,3)\), \(Q=(3,3,3)\), \(R=(1,2,5)\)
\(\overrightarrow{PQ}=(3 - 1,3 - 2,3 - 3)=(2,1,0)\)
\(\overrightarrow{PR}=(1 - 1,2 - 2,5 - 3)=(0,0,2)\)

Step2: Calculate the cross - product \(\overrightarrow{PQ}\times\overrightarrow{PR}\)

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Step3: Find the magnitude of the cross - product

\(\vert\overrightarrow{PQ}\times\overrightarrow{PR}\vert=\sqrt{2^{2}+(-4)^{2}+0^{2}}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\)

Step4: Calculate the area of the triangle

The area of the triangle with two adjacent sides \(\overrightarrow{PQ}\) and \(\overrightarrow{PR}\) is \(A=\frac{1}{2}\vert\overrightarrow{PQ}\times\overrightarrow{PR}\vert\)
\(A=\frac{1}{2}\times2\sqrt{5}=\sqrt{5}\)

Answer:

E. \(\sqrt{5}\)