Scalar Triple Product And Vector Triple Product


Vector Algebra / Thursday, October 31st, 2019

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

\[(1)~~If~~\overrightarrow{\alpha },\overrightarrow{\beta },\overrightarrow{\gamma }~~be~~three~~vectors~~then~~\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]=~~Volume~~of\]

\[the~~parallelopiped~~with~~sides~~\overrightarrow{\alpha },\overrightarrow{\beta }~~and~~\overrightarrow{\gamma }.\]

\[(2)~~\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]=0~~if~~and~~if~~\overrightarrow{\alpha },\overrightarrow{\beta },\overrightarrow{\gamma }~~are~~coplanar.\]

\[(3)~~\frac{1}{6}\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]=~~Volume~~of~~the~~tetrahedron~~with~~sides~~\overrightarrow{\alpha },\overrightarrow{\beta }~~and~~\overrightarrow{\gamma }.\]

\[(4)~~If~~\overrightarrow{\alpha }={{x}_{1}}\hat{i}+{{y}_{1}}\hat{j}+{{z}_{1}}\hat{k},~~\overrightarrow{\beta }={{x}_{2}}\hat{i}+{{y}_{2}}\hat{j}+{{z}_{2}}\hat{k},~~\overrightarrow{\gamma }={{x}_{3}}\hat{i}+{{y}_{3}}\hat{j}+{{z}_{3}}\hat{k}\]

scalar triple product pic

\[(5)~~In~~\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]~~if~~any~~two~~vectors~~are~~\text{interchanged}~~then~~the~~sign\]

\[is~~altered~~but~~the~~value~~remains~~same,~~e.g~~\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]=-\left[ \overrightarrow{\alpha }~~\overrightarrow{\gamma }~~\overrightarrow{\beta } \right]\]

\[(6)~~\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]=0~~if~~any~~two~~vectors~~are~~identical.\]

\[(7)~~If~~any~~of~~the~~vectors~~\overrightarrow{\alpha },\overrightarrow{\beta },\overrightarrow{\gamma }~~is~~sum~~of~~two~~vectors~~then~~is~~\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]\]

\[sum~~of~~two~~triple~~scalar~~product~~e.g~~\left[ \overrightarrow{{{\alpha }_{1}}}+\overrightarrow{{{\alpha }_{2}}}~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]=\left[ \overrightarrow{{{\alpha }_{1}}}~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]+\left[ \overrightarrow{{{\alpha }_{2}}}~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]\]

\[(8)~~If~~any~~of~~the~~vector~~\overrightarrow{\alpha },\overrightarrow{\beta ~}~~and~~\overrightarrow{\gamma }~~is~~a~~scalar~~multiple~~of~~a~~vector\]

\[then~~\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]~~is~~a~~scalar~~multiple~~of~~\left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]~~e.g~~\left[ \overrightarrow{\alpha }~~\lambda \overrightarrow{\beta }~~\overrightarrow{\gamma } \right]=\lambda \left[ \overrightarrow{\alpha }~~\overrightarrow{\beta }~~\overrightarrow{\gamma } \right]\]

\[(9)~~\left[ \hat{i}~~\hat{j}~~\hat{k} \right]=\left[ \hat{j}~~\hat{k}~~\hat{i} \right]=\left[ \hat{k}~~\hat{i}~~\hat{j} \right]=1\]

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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