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Standard Limits to Remove Indeterminate Form

If |a⃗|=5, ...

Question

If $| \to a | = 5, | \to b = 13$ and $| \to a \times \to b | = 25$ , find $\to a \cdot \to b$ .

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Solution

$\begin{matrix} G i v e n, ∣ ∣ \to a ∣ ∣ = 5 ∣ ∣ ∣ \to b ∣ ∣ ∣ = 13 ∣ ∣ ∣ \to a \times \to b ∣ ∣ ∣ = 25 \to a \times \to b = | a | | b | sin θ = 5 \times 13 sin θ = 65 sin θ ∴ sin θ = \frac{25}{65} = \frac{5}{13} b u t w e k n o w t h a t, cos θ = \sqrt{1 - {sin}^{2} θ} = \sqrt{1 - {(\frac{5}{13})}^{2}} = \sqrt{1 - \frac{25}{169}} = \sqrt{\frac{144}{169}} = \frac{12}{13} n o w, \to a . \to b = | a | | b | cos θ = 5 \times 13 \times \frac{12}{13} = 60 s o, \to a . \to b i s 60. \end{matrix}$

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