Name: Vectors: Introduction
Uploaded: 2019-07-19
Duration: 6 min 59 s
Description: Physicists, air traffic controllers, and video game creators have at least one thing in common: vectors. But what exactly are they? Let's dive in to learn more about vectors.

Question

If $\vec{a}, \vec{b} & \vec{c}$ are three non-coplanar vectors, then show that

$\frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} + \frac{\vec{b} \cdot (\vec{a} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} = 0$

Accepted Answer

To prove $\frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} + \frac{\vec{b} \cdot (\vec{a} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} = 0$

Now taking L.H.S we get,

$\frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} + \frac{\vec{b} \cdot (\vec{a} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}}$

Now we know that $[\begin{matrix} \vec{a} & \vec{b} & \vec{c} \end{matrix}] = \vec{a} . (\vec{b} \times \vec{c})$ and $[\begin{matrix} \vec{a} & \vec{b} & \vec{c} \end{matrix}] = [\begin{matrix} \vec{b} & \vec{c} & \vec{a} \end{matrix}] = [\begin{matrix} \vec{c} & \vec{a} & \vec{b} \end{matrix}] = - [\begin{matrix} \vec{b} & \vec{a} & \vec{c} \end{matrix}]$

Now using above formula we get,

$\frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} + \frac{\vec{b} \cdot (\vec{a} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} = \frac{[\begin{matrix} \vec{a} & \vec{b} & \vec{c} \end{matrix}]}{[\begin{matrix} \vec{c} & \vec{a} & \vec{b} \end{matrix}]} + \frac{[\begin{matrix} \vec{b} & \vec{a} & \vec{c} \end{matrix}]}{[\begin{matrix} \vec{c} & \vec{a} & \vec{b} \end{matrix}]}$

$\Rightarrow \frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} + \frac{\vec{b} \cdot (\vec{a} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} = \frac{[\begin{matrix} \vec{a} & \vec{b} & \vec{c} \end{matrix}]}{[\begin{matrix} \vec{c} & \vec{a} & \vec{b} \end{matrix}]} - \frac{[\begin{matrix} \vec{a} & \vec{b} & \vec{c} \end{matrix}]}{[\begin{matrix} \vec{c} & \vec{a} & \vec{b} \end{matrix}]}$

$\Rightarrow \frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} + \frac{\vec{b} \cdot (\vec{a} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} = 0$ which is equal to R.H.S,

Hence proved.

If $\vec{a}, \vec{b} & \vec{c}$ are three non-coplanar vectors, then show that

$\frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} + \frac{\vec{b} \cdot (\vec{a} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} = 0$

Important Questions on Vectors

Learn and Prepare for any exam you want !

If a→, b→ & c→ are three non-coplanar vectors, then show that a→·b→×c→c→×a→·b→+b→·a→×c→c→×a→·b→=0

Important Questions on Vectors

Learn and Prepare for any exam you want !

If $\vec{a}, \vec{b} & \vec{c}$ are three non-coplanar vectors, then show that

$\frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} + \frac{\vec{b} \cdot (\vec{a} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} = 0$