Each-of-the-angle-between-vectors-a-b-and-c-is-equal-to-60-If-a-4-b-2-and-c-6-then-the-modulus-of-a-b-c-is-

Question Number 28640 by sk9236421@gmail.com last updated on 28/Jan/18

Each of the angle between vectors a,

b and c is equal to 60 ° . If ∣ a ∣= 4, ∣ b ∣= 2

and ∣ c ∣= 6, then the modulus of

a + b + c is

Commented by ajfour last updated on 28/Jan/18

Commented by ajfour last updated on 28/Jan/18

Answered by ajfour last updated on 28/Jan/18

l e t \bar{b} = 6 \hat{i} a n d

\bar{a} = 2 \hat{i} + 2 \sqrt{3} \hat{j}

\bar{c} = x \hat{i} + y \hat{j} + z \hat{k}; ∣ \bar{c} ∣= 2

\bar{b} . \bar{c} =∣ \bar{b} ∣∣ \bar{c} ∣ \cos 60 °

\Rightarrow 6 x = 6 \times 2 \times \frac{1}{2}

\Rightarrow x = 1

\bar{a} . \bar{c} =∣ \bar{a} ∣∣ \bar{c} ∣ \cos 60 °

\Rightarrow 2 x + 2 \sqrt{3} y = 4 \times 2 \times \frac{1}{2}

\Rightarrow 2 + 2 \sqrt{3} y = 4

\Rightarrow y = \frac{1}{\sqrt{3}}

a s ∣ \bar{c} ∣= 2 \Rightarrow x^{2} + y^{2} + z^{2} = 4

s o 1 + \frac{1}{3} + z^{2} = 4

\Rightarrow z = \pm \frac{2 \sqrt{2}}{\sqrt{3}}

∣ \bar{a} + \bar{b} + \bar{c} ∣=∣ 6 \hat{i} + 2 \hat{i} + 2 \sqrt{3} \hat{j} + \hat{i} + \frac{1}{\sqrt{3}} \hat{j} \pm \frac{2 \sqrt{2}}{\sqrt{3}} \hat{k} ∣

= \sqrt{81 + \frac{49}{3} + \frac{8}{3}} = \sqrt{81 + 19}

∣ \bar{a} + \bar{b} + \bar{c} ∣= 10 .

Answered by ajfour last updated on 28/Jan/18

∣ \bar{a} + \bar{b} + \bar{c} ∣^{2} = a^{2} + b^{2} + c^{2} + 2 (\bar{a} . \bar{b} + \bar{b} . \bar{c} + \bar{c} . \bar{a})

= 16 + 4 + 36 + (12 + 8 + 24) = 100

∣ \bar{a} + \bar{b} + \bar{c} ∣= 10 .

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