Name: Geometrical Interpretation of Scalar Triple Product
Uploaded: 2019-07-19
Duration: 14 min
Description: Do you know what is the geometrical meaning of the scalar triple product? And why the scalar triple product so important? Check out the video to explore.

Question

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

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

and

C (2, 3, 1) .

Accepted Answer

Let $\vec{a}, \vec{b}, \vec{c}$ be the position vectors of points $A, B$ and $C (2, 3, 1) .$ respectively.

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

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 $A$ .

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

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

$\Rightarrow \vec{A B} = \hat{j} + 2 \hat{k}$

And,

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

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

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

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

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

$\Rightarrow \vec{A B} \times \vec{A C} = - 4 \hat{i} + 2 \hat{j} - \hat{k}$

$\Rightarrow |\vec{A B} \times \vec{A C}| = \sqrt{16 + 4 + 1} = \sqrt{21}$

So, area of $△ A B C = \frac{1}{2} |\vec{A B} \times \vec{A C}| = \frac{\sqrt{21}}{2} sq . units$

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

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

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

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

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