If \overset{\rightarrow}{a} , \textrm{ }\textrm{ } \overset{\rightarrow}{b} \textrm{ }\textrm{ } and \textrm{ }\textrm{ } \overset{\rightarrow}{c} are unit vectors satisfying \left(\left|\right. \overset{\rightarrow}{a} \textrm{ }\textrm{ } - \textrm{ }\textrm{ } \overset{\rightarrow}{b} \left|\right.\right)^{2} \textrm{ }\textrm{ } + \textrm{ }\textrm{ }\textrm{ } \left(\left|\right. \textrm{ } \overset{\rightarrow}{b} \textrm{ }\textrm{ } - \textrm{ }\textrm{ } \overset{\rightarrow}{c} \textrm{ } \left|\right.\right)^{2} \textrm{ }\textrm{ } + \textrm{ }\textrm{ }\textrm{ } \left(\left|\right. \textrm{ } \overset{\rightarrow}{c} \textrm{ }\textrm{ } - \textrm{ }\textrm{ } \overset{\rightarrow}{a} \textrm{ } \left|\right.\right)^{2} \textrm{ } = 9 , then \left|\right. 2 \overset{\rightarrow}{a} \textrm{ }\textrm{ } + \textrm{ }\textrm{ } 5 \overset{\rightarrow}{b} \textrm{ }\textrm{ } + \textrm{ }\textrm{ } 5 \overset{\rightarrow}{c} \textrm{ } \left|\right. is

Engineering

Mathematics

Scalar Product of Two Vectors

Question

If $\vec{a}, \vec{b} and \vec{c}$ are unit vectors satisfying ${| \vec{a} - \vec{b} |}^{2} + {| \vec{b} - \vec{c} |}^{2} + {| \vec{c} - \vec{a} |}^{2} = 9$ , then $| 2 \vec{a} + 5 \vec{b} + 5 \vec{c} |$ is

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${| \vec{a} - \vec{b} |}^{2} + {| \vec{b} - \vec{c} |}^{2} + {| \vec{c} - \vec{a} |}^{2} = 6 - 2 \sum \vec{a} \cdot \vec{b} = 9 \Rightarrow \sum \vec{a} \cdot \vec{b} = \frac{- 3}{2}$

${| \vec{a} + \vec{b} + \vec{c} |}^{2} = \sum {| \vec{a} |}^{2} + 2 \sum \vec{a} \cdot \vec{b} = 0 \Rightarrow \vec{a} + \vec{b} + \vec{c} = \vec{0}$

$\vec{b} + \vec{c} = - \vec{a}$

$| 2 \vec{a} + 5 \vec{b} + 5 \vec{c} | = | 2 \vec{a} - 5 \vec{a} | = 3$