Question
Download Solution PDFThe value of \([ \vec a, \; \vec b + \vec c, \; \vec a + \vec b + \vec c]\) is
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
If \(\vec a, \vec b, \vec c \) are any non-zero vector then
\([ \vec a \; \vec b\; \vec c]= \vec a. ( \vec b \times \vec c) ....(1)\)
Calculation:
Given, \([ \vec a, \; \vec b + \vec c, \; \vec a + \vec b + \vec c]\)
\(= \vec a. [(\vec b + \vec c) \times ( \vec a + \vec b + \vec c)]\)
\(= \vec a. {[ \vec b \times \vec a + \vec b \times \vec b + \vec b \times \vec c + \vec c \times \vec a + \vec c \times \vec b + \vec c \times \vec c]}\)
\(= \vec a. {[ \vec b \times \vec a + \vec b \times \vec c + \vec c \times \vec a + \vec c \times \vec b ]}\)
\(=[ \vec a \; \vec b\; \vec a] + [ \vec a \; \vec b\; \vec c] + [ \vec a \; \vec c\; \vec a]+ [ \vec a \; \vec c\; \vec b]\)
\((since [ \vec a \; \vec b\; \vec a]= 0)\)
\([ \vec a \; \vec b\; \vec c] - [ \vec a \; \vec b\; \vec c]\)
= 0
Last updated on May 6, 2025
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