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

Prove that the three points

\hat{i} + 2 \hat{j} + 3 \hat{k}, - \hat{i} - \hat{j} + 8 \hat{k}

and

- 4 \hat{i} + 4 \hat{j} + 6 \hat{k}

form the vertices of an equilateral triangle.

Accepted Answer

Consider the vertices of the triangle as $\vec{O A} = \hat{i} + 2 \hat{j} + 3 \hat{k}, \vec{O B} = - \hat{i} - \hat{j} + 8 \hat{k}, \vec{O C} = - 4 \hat{i} + 4 \hat{j} + 6 \hat{k}$ . We have to find the sides of the triangle`ABC` by using the above data.

$\vec{A B} = (- \hat{i} - \hat{j} + 8 \hat{k}) - (\hat{i} + 2 \hat{j} + 3 \hat{k}) \Rightarrow - 2 \hat{i} - 3 \hat{j} + 5 \hat{k}$

$\vec{B C} = (- 4 \hat{i} + 4 \hat{j} + 6 \hat{k}) - (- \hat{i} - \hat{j} + 8 \hat{k}) \Rightarrow - 3 \hat{i} + 5 \hat{j} - 2 \hat{k}$

$\vec{C A} = (\hat{i} + 2 \hat{j} + 3 \hat{k}) - (- 4 \hat{i} + 4 \hat{j} + 6 \hat{k}) \Rightarrow 5 \hat{i} - 2 \hat{j} - 3 \hat{k}$

Now finding the length of the sides by taking their magnitude,

$|\vec{A B}| = \sqrt{(- 2)^{2} + (- 3)^{2} + 5^{2}} \Rightarrow \sqrt{4 + 9 + 25} \Rightarrow \sqrt{38}$

$|\vec{B C}| = \sqrt{(- 3)^{2} + 5^{2} + (- 2)^{2}} \Rightarrow \sqrt{9 + 25 + 4} \Rightarrow \sqrt{38}$

$|\vec{C A}| = \sqrt{5^{2} + (- 2)^{2} + (- 3)^{2}} \Rightarrow \sqrt{25 + 4 + 9} \Rightarrow \sqrt{38}$

Since the lengths of all the sides of the triangle are the same i.e., $\sqrt{38}$ , it is an equilateral triangle.

Prove that the three points $\hat{i} + 2 \hat{j} + 3 \hat{k}, - \hat{i} - \hat{j} + 8 \hat{k}$ and $- 4 \hat{i} + 4 \hat{j} + 6 \hat{k}$ form the vertices of an equilateral triangle.

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

Prove that the three points i^+2j^+3k^, -i^-j^+8k^ and -4i^+4j^+6k^ form the vertices of an equilateral triangle.

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

Prove that the three points $\hat{i} + 2 \hat{j} + 3 \hat{k}, - \hat{i} - \hat{j} + 8 \hat{k}$ and $- 4 \hat{i} + 4 \hat{j} + 6 \hat{k}$ form the vertices of an equilateral triangle.