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Applications of Dot Product

If |a| = 2,...

Question

If $| \to a | = 2, | \to b | = 3$ , and $| \to c | = 4$ , then $| \to a - \to b |^{2} + | \to b - \to c |^{2} + | \to c - \to a |^{2} \leq (12 k + λ)$ , then $λ$ is equal to ....................

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Solution

Consider two vectors, $\to a, \to b$
$| \to a - \to b |^{2} = a^{2} + b^{2} - 2 a b c o s θ$

Now this is maximum when $c o s θ = - 1$ or the vectors are anti-parallel.

In that case the above expression reduces to $(a + b)^{2}$ .

Hence the maximum value of
$| \to a - \to b |^{2} + | \to b - \to c |^{2} + \to c - \to a |^{2}$
$= (a + b)^{2} + (b + c)^{2} + (c + a)^{2}$
$= 2 (a^{2} + b^{2} + c^{2}) + 2 (a b + b c + a c)$
$= 2 (4 + 9 + 16) + 2 (6 + 12 + 8)$
$= 2 (29) + 2 (26)$
$= 2 (55)$
$= 110$

Thus
$| \to a - \to b |^{2} + | \to b - \to c |^{2} + | \to c - \to a |^{2} \leq 110$

Now
$12 k + λ = 110$
$110 = 108 + 2$
$= 12 (9) + 2$

Hence $k = 9$ and $λ = 2$ .

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∣ ∣ ∣ ∣ ∣ \begin{matrix} a^{2} + λ^{2} & a b + c λ & c a - b λ a b - c λ & b^{2} + λ^{2} & b c + a λ c a + b λ & b c - a λ & c^{2} + λ^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} λ & c & - b - c & λ & a b & - a & λ \end{matrix} ∣ ∣ ∣ ∣ = {(1 + a^{2} + b^{2} + c^{2})}^{3}

, then the value of

λ

[\to a \times \to b \to b \times \to c \to c \times \to a] = λ [\to a \to b \to c]^{2}

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\to a

\to c

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∣ ∣ ∣ \to b ∣ ∣ ∣ = 4

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\to a

\to c

{cos}^{- 1} (\frac{1}{4})

\to b - 2 \to c = λ \to a

λ

m u = A + \frac{B}{λ} + \frac{C}{λ^{2}}

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∣ ∣ \to a ∣ ∣ = ∣ ∣ \to c ∣ ∣ = 1, ∣ ∣ ∣ \to b ∣ ∣ ∣ = 4, ∣ ∣ ∣ \to b \times \to c ∣ ∣ ∣ = 2

2 \to b = \to c + λ \to a

λ

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