Let us see some more significant properties of the 
:
:
(i) The of three vectors is zero if any two of them are parallel. This implies as a corollary that ![[\vec a\;\;\vec a\;\;\vec b] = 0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sD3YrsWDhNIgSntd9X9_NkFuoYQPsD2oUZPRVJ-_sOAd9yU__DF98k1u7ARjJyl-9emQWPZ8ghu7JlSzzNABvCbu_c_V7-iVXDQMwiPq4gbNFPcRVMHnlEzTDwISSGtrPYg5PPMO0O1pJJNhJ66GX1pZ1ZTViNWV5F8VTpFVy5pAvn7FaEHdt2bC1VzgelMQg=s0-d "[\vec a\;\;\vec a\;\;\vec b] = 0")
(always)
of three vectors is zero if any two of them are parallel. This implies as a corollary that (always)
(ii) For any 
,
,![[\lambda \vec a\;\;\vec b\;\;\vec c] = \lambda [\vec a\;\;\vec b\;\;\vec c]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uH8ReWy3eDIXArYUP1vi0HIFW7JOjj8fmpTQ4EOCwSByzKqG9FRRTF5s4TH05qwsCuS3YqQzkMiDm1N38xE4GsBAWYkj-Du5XSKZSJO34oRxapjfUgv-9rMR7HqKkO5AW1VeqwYIT0KkDRSQ1zZOIPEqaVr_rw-M6Ex7vsXduhCw1jLlNKqe94CnomV0YjMY3pegrOIgN-KcvB4_-dP1idcrNiGTOGBzR7piWNsSPbLPGxfIu_d7nf23Csyptk63VmKnLqyjEKzLWmSYtH8SWzxS3E=s0-d "[\lambda \vec a\;\;\vec b\;\;\vec c] = \lambda [\vec a\;\;\vec b\;\;\vec c]") |
(iii) ![{[(\vec a + \vec b)\,\,\,\,\vec c\;\;\vec d] = [\vec a\;\;\vec c\;\;\vec d] + [\vec b\;\;\vec c\;\;\vec d]}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sWIN0B7WaVoyZYARprwwQOMvh5hbTMHzcVNeQbIRJzKLJdTBLcBDcPjvbug3mkGZmed126TMKvxHFONzXnJZ7j3qGw16crSU_R6F1Jb4RxZYt8C-sFTz2aqVEpfoepiDaH02hgq42pGti4jNL8n1GtZXVgEKRm9IAiGQgZIKfjMlOWsjLGTgGhIiwKBr99qZJ14krWIezQooqzIOKlIWICRd44C1NVTnVpMMVWmd9ry-ab7UGghHJMBnEaKqCKAD9WzPkf4V966tHjAmQoIDbvk58oZSVZWgn8Rt5QsdlEjE2jotAdabAgiCSNfS6T8TkqgAQjUVpiIy6ef5h99LViUnefl3AEAo2VWNxX_Is=s0-d "{[(\vec a + \vec b)\,\,\,\,\vec c\;\;\vec d] = [\vec a\;\;\vec c\;\;\vec d] + [\vec b\;\;\vec c\;\;\vec d]}")
This property is very important and is used extremely frequently. The justification is straight forward:
![[(\vec a + \vec b)\;\vec c\;\;\vec d] = (\vec a + \vec b) \cdot (\vec c \times \vec d)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vctB0je1fRkcvdDSK31481mSc4G4m52n269m3SwCazpwVW4w0WWV4UrtZhtXwf8P2UkrSEOiAdKX2kBKLCVUFYZl0SSUxHIfFy79yB902I3YAj94zvEUxEvBZlSHQARWY9xUyzhIDSr0CXKZEDKwaEy1HvnPlEF8F_T0CK2Nea2sxNMNCg-t2r-cGrVBO57ovPseFmeKLtIRdvjghSRq6bAJ3_R0cgh6CUunomUDNYwgEeonCqvFv1ly9z4cqb3ONaXYUhVvTVTCEJ3kdKObBgPOCRw_sE1dU=s0-d "[(\vec a + \vec b)\;\vec c\;\;\vec d] = (\vec a + \vec b) \cdot (\vec c \times \vec d)") | |
 Dot product is distributive over addition | |
![= [\vec a\;\;\vec c\;\;\vec d] + [\vec b\;\;\vec c\;\;\vec d]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uTwTE4kJ3EMy-4MPbD8MJJyRaGu9MCN-SA3IBg5OSUXEt63cWdSPm2tF-Yu4gKfCQlCguTA5J5jQrPLJ24UQVAhbl40R4J8SESF4UEy8x0K2be5WyYdZALfJ3hOuEiz87SiF06rQhVPP79i7G7l7n7fh98TE4TlaQKZqmlEWWZSShJ_nJC1js2lzieI26lXYF-RfGU16DAdaQpxsnIqlyr6z0F3Hluc2CSffe2vppb3w9hecvLpJIxYZYqsnYAweoZJa6W=s0-d "= [\vec a\;\;\vec c\;\;\vec d] + [\vec b\;\;\vec c\;\;\vec d]") |
(iv) Three vector are coplanar if and only if their is zero. This is because the volume of the parallelopiped formed by the three vectors becomes zero if they are coplanar.
is zero. This is because the volume of the parallelopiped formed by the three vectors becomes zero if they are coplanar.
You are urged to rigorously prove the other way implication, i.e, prove that if ![[\vec a\;\;\vec b\;\;\vec c] = 0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_um9MBEqZzqa1Ydytj2Gx73OCuFianQsLBYq3rDAma-_ANnyfuGFJcW6im0jBVECRvClNdb5ToET3njVh-c7wy62AqyLKnd9PwLvvMVblooHWIj8pQu7sCSULABF4cNJ0QA6_9G3Aj9iPFbjzSb3lEmFKa2aVPNmfknKfwkkIrg7hWfewyn4dK9dG0k8X_a-5F5=s0-d "[\vec a\;\;\vec b\;\;\vec c] = 0")
where 
are non-zero non-collinear vectors, then 
must be coplanar.
where are non-zero non-collinear vectors, then must be coplanar.
(v) Let 
and 
. Then,
and . Then,![[\vec a\;\;\vec b\;\;\vec c] = \vec a \cdot (\vec b \times \vec c) = ({a_1}\hat i + {a_2}\hat j + {a_3}\hat k) \cdot \left| {\begin{array}{*{20}{c}} {\hat i}\,\,\,\,{\hat j}\,\,\,\,{\hat k}\\ {{b_1}}\,\,\,\,{{b_2}}\,\,\,\,{{b_3}}\\ {{c_1}}\,\,\,\,{{c_2}}\,\,\,\,{{c_3}} \end{array}} \right|](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u8pJuxkO12kzb3WDY3NPqjzLbK8aBk2dm-d_0jHoVDDVC4TSh-80UsSU4BlZWnZ09ku3pK8pD35eytU75lfCLGcUO2qc8kQBMfElre87d7HKlMTkBSyxKej7fb0dpJRhEBZC6fzF1vwnwN_-wxa8Ps9weLWY25hHx6UVbxGuucERHlH0hBUnMrtWljsD-ieJwAklOSuMCEdsSpxJO_B3XqEOsLkNUpfknCtQBW_Zz7u8Z7gY0uJNNa0UZGmgGlYFL-3Sm9_2u4777h_H-0NzZD9JTsrfhVyxQ9pkCQ4tqiYwg-jDkvJ63AVrj6lNVmYPvJHE_8l6C8vMXSUFHMfQv5ojs_fff8m9b-ImCXU1c5VHe65hWzSbdGII7S7KgxX09_imdVjFVKMOWvvmnEeoC9XDzNZLF9K2I0UUnhpPzguWUQkIeAc9hEht5FtO9u5ug5vQXs-zQZjpCLhXNYIm62R-CwpQUp16VK1ssAUI1olNg6DqOrwybEoF5ekAqjW3aWxX_VzmY_aZMpenlwePbjIVX1Yyc_VsuZY8Ic-64ynIiWUeDdq5EAQlI2RCwClvaEoF3CaCJznIsOtoYwo7uy36q6jAa61WSV3yXHPXsC09PHQ7k1BO1rh_krSfjkXxc8FW_YPuFX6lWaNIFt8eA5JN7brz-TkmageEE3w3aSRo7x5hzOrqTUAN620AotqhhXdSUwdkXxll-PtNrW3dCu25XVid_ypTo-bIavwklV93h_h5-urbkRB35BdJMqEP2p-RZGaayxpGOU7xGdYMHyBhtMLnu62u1vKA=s0-d "[\vec a\;\;\vec b\;\;\vec c] = \vec a \cdot (\vec b \times \vec c) = ({a_1}\hat i + {a_2}\hat j + {a_3}\hat k) \cdot \left| {\begin{array}{*{20}{c}} {\hat i}\,\,\,\,{\hat j}\,\,\,\,{\hat k}\\ {{b_1}}\,\,\,\,{{b_2}}\,\,\,\,{{b_3}}\\ {{c_1}}\,\,\,\,{{c_2}}\,\,\,\,{{c_3}} \end{array}} \right|") | |
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This relation is quite useful and is worth remembering.
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