Using vectors, find the area of triangle with vertices $A (2, 3, 5), B (3, 5, 8), C (2, 7, 8)$ .

Open in App

Solution

Let $- - \to O A, - - \to O B, - - \to O C$ be the position vectors of points A, B and C respectively.
Then, $- - \to O A = 2 ˆ i + 3 ˆ j + 5 ˆ k, - - \to O B = 3 ˆ i + 5 ˆ j + 8 ˆ k, - - \to O C = 2 ˆ i + 7 ˆ j + 8 ˆ k$
Then, $- - \to A B = - - \to O B - - - \to O A = ˆ i + 2 ˆ j + 3 ˆ k$
and, $- - \to A C = - - \to O C - - - \to O A = 4 ˆ j + 3 ˆ k$

Now, area of $△ A B C = | \frac{1}{2} ∣ ∣ ∣ ∣ ∣ \begin{matrix} ˆ i & ˆ j & ˆ k 1 & 2 & 3 0 & 4 & 3 \end{matrix} ∣ ∣ ∣ ∣ ∣ |$

$= \frac{1}{2} | - 6 ˆ i - 3 ˆ j + 4 ˆ k |$

$= \frac{1}{2} \sqrt{36 + 9 + 16} = \frac{1}{2} \sqrt{61} s q . u n i t s$

Suggest Corrections

Similar questions

Using vectors, find the area of the $△$ ABC with vertices A(1, 2, 3), B (2, -1, 4) and C(4, 5, -1).

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

(1, 5, 5)

Join BYJU'S Learning Program