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

The position vectors of two points

A

and

B

are

\hat{i} + \hat{j} + \hat{k}

and

5 \hat{i} - 3 \hat{j} + \hat{k}

. Find a unit vector in the direction of

\vec{A B}

and also find the directional cosines of

\vec{A B}

. What angles does

\vec{A B}

make with the three axes?

Accepted Answer

The position vectors of $\vec{A}$ and $\vec{B}$ are given as $\hat{i} + \hat{j} + \hat{k}$ and $5 \hat{i} - 3 \hat{j} + \hat{k}$ .

$\vec{A B} = \vec{O B} - \vec{O A}$

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

$= (5 - 1) \hat{i} + (- 3 - 1) \hat{j} + (1 - 1) \hat{k}$

$B$

Unit vector in the direction of $\vec{A B}$ can be found out as $\vec{u_{A B}} = \frac{\vec{A B}}{|\vec{A B}|}$ .

$\vec{u_{A B}} = \frac{(4 \hat{i} - 4 \hat{j})}{\sqrt{4^{2} + 4^{2}}} \Rightarrow \frac{4 (\hat{i} - \hat{j})}{\sqrt{32}} \Rightarrow \frac{(\hat{i} - \hat{j})}{\sqrt{2}} \Rightarrow \frac{\hat{i}}{\sqrt{2}} - \frac{\hat{j}}{\sqrt{2}}$

The directional cosines are the coefficients of the unit vector of $\vec{A B} = \frac{1}{\sqrt{2}}, \frac{- 1}{\sqrt{2}}, 0$ .

Directional cosines $l, m$ and $n$ of a vector are the angles $\cos α, \cos β$ and $\cos γ$ that the vector makes with the $x, y$ and $z$ axes respectively.

So, $\cos α = \frac{1}{\sqrt{2}}, \cos β = \frac{- 1}{\sqrt{2}}, \cos γ = 0$ .

The angle made by $\vec{A B}$ with the $x - axis$ is given by $α = \cos^{- 1} (\frac{1}{\sqrt{2}}) = 45 °$ .

The angle made by $\vec{A B}$ with the $y - axis$ is given by $β = \cos^{- 1} (\frac{- 1}{\sqrt{2}}) = 135 °$ .

The angle made by $\vec{A B}$ with the $z - axis$ is given by $γ = \cos^{- 1} (0) = 90 °$ .

The position vectors of two points $A$ and $B$ are $\hat{i} + \hat{j} + \hat{k}$ and $5 \hat{i} - 3 \hat{j} + \hat{k}$ . Find a unit vector in the direction of $\vec{A B}$ and also find the directional cosines of $\vec{A B}$ . What angles does $\vec{A B}$ make with the three axes?

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

The position vectors of two points A and B are i^+j^+k^ and 5i^-3j^+k^. Find a unit vector in the direction of AB→ and also find the directional cosines of AB→. What angles does AB→ make with the three axes?

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

The position vectors of two points $A$ and $B$ are $\hat{i} + \hat{j} + \hat{k}$ and $5 \hat{i} - 3 \hat{j} + \hat{k}$ . Find a unit vector in the direction of $\vec{A B}$ and also find the directional cosines of $\vec{A B}$ . What angles does $\vec{A B}$ make with the three axes?