Let a¯, b¯, c¯ be three vectors such that the magnitude of b¯ is twice that of a¯ and magnitude of c¯is three times that of a¯ . If the angle between each pair of vectors is π3 and |a¯+b¯+c¯|=5 , then |c¯|+|a¯|+|b¯|=

Given, |b^|=2|a^| and |c^|=3|a^|Angle between each pair of vector is π3. |a^+b^+c^|2=|a^|2+|b^|2+|c^|2+2|a^||b^| cosπ3+2|b^||c^| cosπ3+2|a^||c^| cosπ3 25=|a^|2+4|a^|2+9|a^|2+4|a^|2×12+2×6|a^|2×12+6|a^|2×12 25=25|a^|2 ⇒ |a^|=1 ∴ |b^|=2 ⇒ |c^|=3 ⇒|a^|+|b^|+|c^|=1+2+3=6

Let a¯, b¯, c¯ be three vectors such that the magnitude of b¯ is twice that of a¯ and magnitude of c¯is three times that of a¯ . If the angle between each pair of vectors is π3 and |a¯+b¯+c¯|=5 , then |c¯|+|a¯|+|b¯|=

Given, |b^|=2|a^| and |c^|=3|a^|Angle between each pair of vector is π3. |a^+b^+c^|2=|a^|2+|b^|2+|c^|2+2|a^||b^| cosπ3+2|b^||c^| cosπ3+2|a^||c^| cosπ3 25=|a^|2+4|a^|2+9|a^|2+4|a^|2×12+2×6|a^|2×12+6|a^|2×12 25=25|a^|2 ⇒ |a^|=1 ∴ |b^|=2 ⇒ |c^|=3 ⇒|a^|+|b^|+|c^|=1+2+3=6

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Let $\bar{a}, \bar{b}, \bar{c}$ be three vectors such that the magnitude of $\bar{b}$ is twice that of $\bar{a}$ and magnitude of $\bar{c}$ is three times that of $\bar{a}$ . If the angle between each pair of vectors is $\frac{π}{3}$ and $| \bar{a} + \bar{b} + \bar{c} | = 5$ , then $| \bar{c} | + | \bar{a} | + | \bar{b} | =$

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$3 \sqrt{2}$

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Detailed Solution

Given, $| \hat{b} | = 2 | \hat{a} |$ and $| \hat{c} | = 3 | \hat{a} |$
Angle between each pair of vector is $\frac{π}{3}$ .
${| \hat{a} + \hat{b} + \hat{c} |}^{2} = {| \hat{a} |}^{2} + {| \hat{b} |}^{2} + {| \hat{c} |}^{2} + 2 | \hat{a} | | \hat{b} | \cos \frac{π}{3} + 2 | \hat{b} | | \hat{c} | \cos \frac{π}{3} + 2 | \hat{a} | | \hat{c} | \cos \frac{π}{3} 25 = {| \hat{a} |}^{2} + 4 {| \hat{a} |}^{2} + 9 {| \hat{a} |}^{2} + 4 {| \hat{a} |}^{2} \times \frac{1}{2} + 2 \times 6 {| \hat{a} |}^{2} \times \frac{1}{2} + 6 {| \hat{a} |}^{2} \times \frac{1}{2} 25 = 25 {| \hat{a} |}^{2} \Rightarrow | \hat{a} | = 1 ∴ | \hat{b} | = 2 \Rightarrow | \hat{c} | = 3 \Rightarrow | \hat{a} | + | \hat{b} | + | \hat{c} | = 1 + 2 + 3 = 6$