If ${ˆ a}_{1} + {ˆ a}_{2} + {ˆ a}_{3} = 0$ $hata_1+hata_2+hata_3=0$ then the value of ${| {ˆ a}_{1} - {ˆ a}_{2} |}_{1}^{2} + {| {ˆ a}_{2} - {ˆ a}_{3} |}_{2}^{2} + {| {ˆ a}_{3} - {ˆ a}_{1} |}_{3}^{2}$ $|hata_1-hata_2|^2+|hata_2-hata_3|^2+|hata_3-hata_1|^2$ is?

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If ${ˆ a}_{1} + {ˆ a}_{2} + {ˆ a}_{3} = 0$ $hata_1+hata_2+hata_3=0$ then the value of ${| {ˆ a}_{1} - {ˆ a}_{2} |}_{1}^{2} + {| {ˆ a}_{2} - {ˆ a}_{3} |}_{2}^{2} + {| {ˆ a}_{3} - {ˆ a}_{1} |}_{3}^{2}$ $|hata_1-hata_2|^2+|hata_2-hata_3|^2+|hata_3-hata_1|^2$ is?

options are

6

9

4

8

1 Answer

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Answer 1 · 2018-06-22T00:33:49

9

Explanation:

${ˆ a}_{1} + {ˆ a}_{2} + {ˆ a}_{3} = 0 \Rightarrow$ $hat a_1+hat a_2+hat a_3=0 implies$

${({ˆ a}_{1} + {ˆ a}_{2} + {ˆ a}_{3})}_{1}^{2} = 0 \Rightarrow$ $(hat a_1+hat a_2+hat a_3)^2=0 implies$

${| {ˆ a}_{1} |}_{1}^{2} + {| {ˆ a}_{2} |}_{2}^{2} + {| {ˆ a}_{3} |}_{3}^{2} + 2 {ˆ a}_{1} \cdot {ˆ a}_{2} + 2 {ˆ a}_{2} \cdot {ˆ a}_{3} + 2 {ˆ a}_{3} \cdot {ˆ a}_{1} = 0$ $|hat a_1|^2+|hat a_2|^2+|hat a_3|^2+2hat a_1 cdot hat a_2+2hat a_2 cdot hat a_3+2hat a_3 cdot hat a_1=0$

Since the vectors are all of unit length, we have

$2 {ˆ a}_{1} \cdot {ˆ a}_{2} + 2 {ˆ a}_{2} \cdot {ˆ a}_{3} + 2 {ˆ a}_{3} \cdot {ˆ a}_{1} = - 3$ $2hat a_1 cdot hat a_2+2hat a_2 cdot hat a_3+2hat a_3 cdot hat a_1 = -3$

Now

${| {ˆ a}_{1} - {ˆ a}_{2} |}_{1}^{2} + {| {ˆ a}_{2} - {ˆ a}_{3} |}_{2}^{2} + {| {ˆ a}_{3} - {ˆ a}_{1} |}_{3}^{2}$ $|hat a_1-hat a_2|^2+|hat a_2-hat a_3|^2+|hat a_3-hat a_1|^2$
$= ({| {ˆ a}_{1} |}_{1}^{2} + {| {ˆ a}_{2} |}_{2}^{2} - 2 {ˆ a}_{1} \cdot {ˆ a}_{2})$ $= (|hat a_1|^2+|hat a_2|^2-2hat a_1 cdot hat a_2)$
$qquad + (|hat a_2|^2+|hat a_3|^2-2hat a_2 cdot hat a_3)$
$qquad + (|hat a_3|^2+|hat a_1|^2-2hat a_3 cdot hat a_1)$
$= 6-(2hat a_1 cdot hat a_2+2hat a_2 cdot hat a_3+2hat a_3 cdot hat a_1)$
$=6-(-3) = 9$

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If ${ˆ a}_{1} + {ˆ a}_{2} + {ˆ a}_{3} = 0$ $hata_1+hata_2+hata_3=0$ then the value of ${| {ˆ a}_{1} - {ˆ a}_{2} |}_{1}^{2} + {| {ˆ a}_{2} - {ˆ a}_{3} |}_{2}^{2} + {| {ˆ a}_{3} - {ˆ a}_{1} |}_{3}^{2}$ $|hata_1-hata_2|^2+|hata_2-hata_3|^2+|hata_3-hata_1|^2$ is?

options are

6

9

4

8

1 Answer

Explanation:

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Science

Math

Humanities

... and beyond

If hata_1+hata_2+hata_3=0ˆa1+ˆa2+ˆa3=0hata_1+hata_2+hata_3=0 then the value of |hata_1-hata_2|^2+|hata_2-hata_3|^2+|hata_3-hata_1|^2|ˆa1−ˆa2|2+|ˆa2−ˆa3|2+|ˆa3−ˆa1|2|hata_1-hata_2|^2+|hata_2-hata_3|^2+|hata_3-hata_1|^2 is?

options are 6 9 4 8

1 Answer

Explanation:

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Impact of this question

If ${ˆ a}_{1} + {ˆ a}_{2} + {ˆ a}_{3} = 0$ $hata_1+hata_2+hata_3=0$ then the value of ${| {ˆ a}_{1} - {ˆ a}_{2} |}_{1}^{2} + {| {ˆ a}_{2} - {ˆ a}_{3} |}_{2}^{2} + {| {ˆ a}_{3} - {ˆ a}_{1} |}_{3}^{2}$ $|hata_1-hata_2|^2+|hata_2-hata_3|^2+|hata_3-hata_1|^2$ is?

options are

6

9

4

8