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 straight line joining the mid-points of two non-parallel sides of a trapezium is parallel to the parallel sides and is half of their sum.

Accepted Answer

Consider a trapezium $P Q R S$ . Let $T U$ be the line segment joining the mid-points of two non-parallel sides $P Q$ and $R S$ . Let the position vectors of points $P, Q, R$ and $S$ be $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ , respectively. Draw the figure for the data as below.

We have to prove that $T U$ is parallel to $P S$ as well as $Q R$ . For that, we need to find the position vectors of $T$ and $U$ first. $T$ is the mid-point of $Q P$ and $U$ is the mid-point of $R S$ .

$\vec{T} = \frac{\vec{a} + \vec{b}}{2}$ and $\vec{U} = \frac{\vec{c} + \vec{d}}{2}$

Computing the vectors $P S$ and $Q R$ ,

$\vec{P S} = \vec{d} - \vec{a} \dots \dots \dots \dots \dots (1)$

$\vec{Q R} = \vec{c} - \vec{b} \dots \dots \dots \dots \dots (2)$

Now computing vector $T U$ ,

$\vec{T U} = \frac{\vec{c} + \vec{d}}{2} - \frac{\vec{a} + \vec{b}}{2}$

$= \frac{1}{2} (\vec{c} + \vec{d} - \vec{a} - \vec{b})$

$= \frac{1}{2} [(\vec{c} - \vec{b}) + (\vec{d} - \vec{a})]$

From equations $1$ and $2$ , we can substitute the values in the above equation,

$\vec{T U} = \frac{1}{2} [\vec{Q R} + \vec{P S}]$

$= \frac{1}{2} \vec{Q R} + \frac{1}{2} \vec{P S}$

As we know $\vec{P S} | | \vec{Q R}, \vec{P S} = λ \vec{Q R}$ .

Hence, $\vec{T U}$ is parallel to the parallel sides.

Also, it is clear that $T U$ is equal to half of the sum of the parallel sides.

Prove that the straight line joining the mid-points of two non-parallel sides of a trapezium is parallel to the parallel sides and is half of their sum.

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