Name: Scalar Triple Product
Uploaded: 2019-07-19
Duration: 6 min 56 s
Description: Have you ever heard about scalar triple products? They involve both dot and cross products! Interesting, aren't they? Watch this amazing video to learn more...

Question

Using vectors, find the area of the triangle whose vertices are:

$A (3, - 1, 2), B (1, - 1, - 3) and C (4, - 3, 1)$ .

Accepted Answer

Let $A (3, - 1, 2), B (1, - 1, - 3) and C (4, - 3, 1)$ be the position vectors of points $A, B$ and $C$ respectively.

Then, $\vec{a} = 3 \hat{i} - \hat{j} + 2 \hat{k}, \vec{b} = \hat{i} - \hat{j} - 3 \hat{k}, \vec{c} = 4 \hat{i} - 3 \hat{j} + \hat{k}$ .

Now, area of $△ A B C = \frac{1}{2} |\vec{A B} \times \vec{A C}|$

$\vec{A B} =$ Position vector of $B -$ Position vector of $A$ .

$\Rightarrow \vec{A B} = \vec{b} - \vec{a}$

$\Rightarrow \vec{A B} = (\hat{i} - \hat{j} - 3 \hat{k}) - (3 \hat{i} - \hat{j} + 2 \hat{k})$

$\Rightarrow \vec{A B} = - 2 \hat{i} - 5 \hat{k}$

and $\vec{A C} =$ Position vector of $C -$ Position vector of $A$ .

$\Rightarrow \vec{A C} = \vec{c} - \vec{a}$

$\Rightarrow \vec{A C} = (4 \hat{i} - 3 \hat{j} + \hat{k}) - (3 \hat{i} - \hat{j} + 2 \hat{k})$

$\Rightarrow \vec{A C} = \hat{i} - 2 \hat{j} - \hat{k}$

$∴ \vec{A B} \times \vec{A C} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - 2 & 0 & - 5 \\ 1 & - 2 & - 1 \end{matrix}|$

$= \hat{i} (- 10) - \hat{j} (2 + 5) + \hat{k} (4)$

$= - 10 \hat{i} - 7 \hat{j} + 4 \hat{k}$

$\Rightarrow |\vec{A B} \times \vec{A C}| = \sqrt{(- 10)^{2} + (- 7)^{2} + 4^{2}} = \sqrt{165}$

So, area of $△ A B C = \frac{1}{2} |\vec{A B} \times \vec{A C}| = \frac{\sqrt{165}}{2} sq . units$

Using vectors, find the area of the triangle whose vertices are:

$A (3, - 1, 2), B (1, - 1, - 3) and C (4, - 3, 1)$ .

Important Questions on Vectors

Learn and Prepare for any exam you want !

Using vectors, find the area of the triangle whose vertices are: A(3,−1,2),B(1,−1,−3) and C(4,−3,1).

Important Questions on Vectors

Learn and Prepare for any exam you want !

Using vectors, find the area of the triangle whose vertices are:

$A (3, - 1, 2), B (1, - 1, - 3) and C (4, - 3, 1)$ .