The area of triangle whose vertices are \left(\right. 1 , 2 , 3 \left.\right) , \left(\right. 2 , 5 , - 1 \left.\right) and \left(\right.

Engineering

Mathematics

Line of Greatest Slope

Question

The area of triangle whose vertices are

(1, 2, 3), (2, 5, - 1)

and

(- 1, 1, 2)

150 s q . u n i t s

145 s q . u n i t s

\sqrt{155} / 2 s q . u n i t s

155 / 2 s q . u n i t s

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Let the vertices of triangle are

A (1, 2, 3), B (2, 5, - 1)

and

C (- 1, 1, 2)

Then,

\begin{array}{l} \vec{A B} = \vec{O B} - \vec{O A} = \hat{i} + 3 \hat{j} - 4 \hat{k}, \\ , \\ \vec{A C} = \vec{O C} - \vec{O A} = - 2 \hat{i} - \hat{j} - \hat{k}, \end{array}

Then,

\begin{array}{l} \overrightarrow { AB } \times \overrightarrow { AC } =\left| { \begin{array} { *{ 20 }{ c } }{ \hat { i },} &amp; { \hat { j },} &amp; { \hat { k },} \\ 1 &amp; 3 &amp; { -4 } \\ { -2 } &amp; { -1 } &amp; { -1 } \end{array} } \right|,\\,\\ =-7\hat { i } +9\hat { j } +5\hat { k },\\,\\ \left| { \overrightarrow { AB } \times \overrightarrow { AC },} \right| =\sqrt { { { \left( { -7 } \right),}^{ 2 } }+{ { \left( 9 \right),}^{ 2 } }+{ { \left( 5 \right),}^{ 2 } } },\\,\\ =\sqrt { 49+8+25 },\\,\\ =\sqrt { 155 },\end{array}

Area of triangle

A B C = \frac{1}{2} ∣ \vec{A B} \times \vec{A C} ∣

= \frac{1}{2} \times \sqrt{155}

= \frac{\sqrt{155}}{2}

sq. units

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