Table of Contents
Scalar Triple Product
If α, β and γ be three vectors then the product (α X β) . γ is called triple scalar product (or, box product) of . It is denoted by [ α β γ].
Note: [ α β γ] is a scalar quantity.
Properties of Triple Scalar Product
\[(1)~~If~~\overrightarrow{\alpha },\overrightarrow{\beta },\overrightarrow{\gamma }~~be~~three~~vectors~~then~~\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]=~~Volume~~of\]
\[the~~parallelopiped~~with~~sides~~\overrightarrow{\alpha },\overrightarrow{\beta }~~and~~\overrightarrow{\gamma }.\]
\[(2)~~\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]=0~~if~~and~~if~~\overrightarrow{\alpha },\overrightarrow{\beta },\overrightarrow{\gamma }~~are~~coplanar.\]
\[(3)~~\frac{1}{6}\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]=~~Volume~~of~~the~~tetrahedron~~with~~sides~~\overrightarrow{\alpha },\overrightarrow{\beta }~~and~~\overrightarrow{\gamma }.\]
\[(4)~~If~~\overrightarrow{\alpha }={{x}_{1}}\hat{i}+{{y}_{1}}\hat{j}+{{z}_{1}}\hat{k},~~\overrightarrow{\beta }={{x}_{2}}\hat{i}+{{y}_{2}}\hat{j}+{{z}_{2}}\hat{k},~~\overrightarrow{\gamma }={{x}_{3}}\hat{i}+{{y}_{3}}\hat{j}+{{z}_{3}}\hat{k}\]
\[(5)~~In~~\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]~~if~~any~~two~~vectors~~are~~\text{interchanged}~~then~~the~~sign\]
\[is~~altered~~but~~the~~value~~remains~~same,~~e.g~~\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]=-\left[ \overrightarrow{\alpha }~~\overrightarrow{\gamma }~~\overrightarrow{\beta } \right]\]
\[(6)~~\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]=0~~if~~any~~two~~vectors~~are~~identical.\]
\[(7)~~If~~any~~of~~the~~vectors~~\overrightarrow{\alpha },\overrightarrow{\beta },\overrightarrow{\gamma }~~is~~sum~~of~~two~~vectors~~then~~is~~\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]\]
\[sum~~of~~two~~triple~~scalar~~product~~e.g~~\left[ \overrightarrow{{{\alpha }_{1}}}+\overrightarrow{{{\alpha }_{2}}}~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]=\left[ \overrightarrow{{{\alpha }_{1}}}~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]+\left[ \overrightarrow{{{\alpha }_{2}}}~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]\]
\[(8)~~If~~any~~of~~the~~vector~~\overrightarrow{\alpha },\overrightarrow{\beta ~}~~and~~\overrightarrow{\gamma }~~is~~a~~scalar~~multiple~~of~~a~~vector\]
\[then~~\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]~~is~~a~~scalar~~multiple~~of~~\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]~~e.g~~\left[ \overrightarrow{\alpha }~~\lambda \overrightarrow{\beta }~~\overrightarrow{\gamma } \right]=\lambda \left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]\]
\[(9)~~\left[ \hat{i}~~\hat{j}~~\hat{k} \right]=\left[ \hat{j}~~\hat{k}~~\hat{i} \right]=\left[ \hat{k}~~\hat{i}~~\hat{j} \right]=1\]
Vector Triple Product
If α, β and γ be three vectors then the product
\[\left( \overrightarrow{\alpha }\times \overrightarrow{\beta } \right)\times \overrightarrow{\gamma }\]
is called vector triple product of α, β and γ.
Properties of Vector Triple Product
\[(1)~~\left( \overrightarrow{\alpha }\times \overrightarrow{\beta } \right)\times \overrightarrow{\gamma }=\left( \overrightarrow{\alpha }.\overrightarrow{\gamma } \right)\overrightarrow{\beta }-\left( \overrightarrow{\beta }.\overrightarrow{\gamma } \right)\overrightarrow{\alpha }\]
\[(2)~~\overrightarrow{\alpha }\times \left( \overrightarrow{\beta }\times \overrightarrow{\gamma } \right)=\left( \overrightarrow{\alpha }.\overrightarrow{\gamma } \right)\overrightarrow{\beta }-\left( \overrightarrow{\alpha }.\overrightarrow{\beta } \right)\overrightarrow{\gamma }\]
\[(3)~~\left( \overrightarrow{\alpha }\times \overrightarrow{\beta } \right)\times \overrightarrow{\gamma }\ne \overrightarrow{\alpha }\times \left( \overrightarrow{\beta }\times \overrightarrow{\gamma } \right)\]
Associative law for cross product fails.
| Example 01 |
Prove that
\[\left( \overrightarrow{\alpha }\times \overrightarrow{\beta } \right).\left( \overrightarrow{\gamma }\times \overrightarrow{\delta } \right)=\left( \overrightarrow{\alpha }.\overrightarrow{\gamma } \right)\left( \overrightarrow{\beta }.\overrightarrow{\delta } \right)-\left( \overrightarrow{\alpha }.\overrightarrow{\delta } \right)\left( \overrightarrow{\beta }.\overrightarrow{\gamma } \right)\]
Solution:
\[L.H.S=\left( \overrightarrow{\alpha }\times \overrightarrow{\beta } \right).\left( \overrightarrow{\gamma }\times \overrightarrow{\delta } \right)=\left( \overrightarrow{\alpha }\times \overrightarrow{\beta } \right).\overrightarrow{\rho }~~where~~\overrightarrow{\rho }=\left( \overrightarrow{\gamma }\times \overrightarrow{\delta } \right)\]
\[=\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\rho } \right]=-\left[ \overrightarrow{\alpha }~~\overrightarrow{\rho }~~\overrightarrow{\beta } \right]~=-\left( \overrightarrow{\alpha }\times \overrightarrow{\rho } \right).\overrightarrow{\beta }\]
\[=-\left\{ \overrightarrow{\alpha }\times \left( \overrightarrow{\gamma }\times \overrightarrow{\delta } \right) \right\}.\overrightarrow{\beta }\]
\[=-\left\{ \left( \overrightarrow{\alpha }.\overrightarrow{\delta } \right)\overrightarrow{\gamma }-\left( \overrightarrow{\alpha }.\overrightarrow{\gamma } \right)\overrightarrow{\delta } \right\}.\overrightarrow{\beta }\]
\[=\left\{ \left( \overrightarrow{\alpha }.\overrightarrow{\gamma } \right)\overrightarrow{\delta } \right\}.\overrightarrow{\beta }-\left\{ \left( \overrightarrow{\alpha }.\overrightarrow{\delta } \right)\overrightarrow{\gamma } \right\}.\overrightarrow{\beta }\]
\[=\left( \overrightarrow{\alpha }.\overrightarrow{\gamma } \right)\left( \overrightarrow{\delta }.\overrightarrow{\beta } \right)-\left( \overrightarrow{\alpha }.\overrightarrow{\gamma } \right)\left( \overrightarrow{\gamma }.\overrightarrow{\beta } \right)\]
\[=\left( \overrightarrow{\alpha }.\overrightarrow{\gamma } \right)\left( \overrightarrow{\beta }.\overrightarrow{\delta } \right)-\left( \overrightarrow{\alpha }.\overrightarrow{\delta } \right)\left( \overrightarrow{\beta }.\overrightarrow{\gamma } \right)=R.H.S\]
| Example 02 |
Prove that
\[\left( \overrightarrow{\alpha }\times \overrightarrow{\beta } \right)\times \left( \overrightarrow{\gamma }\times \overrightarrow{\delta } \right)=\overrightarrow{\beta }\left[ \overrightarrow{\gamma }~~\overrightarrow{\delta }~~\overrightarrow{\alpha } \right]-\overrightarrow{\alpha }\left[ \overrightarrow{\gamma ~}~~\overrightarrow{\delta }~~\overrightarrow{\beta } \right]\]
Hence prove that if the α, β, γ, δ are coplanar vectors
\[\left( \overrightarrow{\alpha }\times \overrightarrow{\beta } \right)\times \left( \overrightarrow{\gamma }\times \overrightarrow{\delta } \right)=\overrightarrow{0}\]
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