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Applications of Dot Product

If | a⃗| = ...

Question

If $| \to a | = \sqrt{26}, ∣ ∣ \to b ∣ ∣ = 7$ and $∣ ∣ \to a \times \to b ∣ ∣ = 35$ , Find $\to a . \to b$

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Solution

We have

$∣ ∣ \to a ∣ ∣ = \sqrt{26}$

$∣ ∣ ∣ \to b ∣ ∣ ∣ = 7$

$∣ ∣ ∣ \to a \times \to b ∣ ∣ ∣ = 35$

We know that

$∣ ∣ ∣ \to A \times \to B ∣ ∣ ∣ = ∣ ∣ ∣ \to A ∣ ∣ ∣ ∣ ∣ ∣ \to B ∣ ∣ ∣ sin θ$

So,

$∣ ∣ ∣ \to a \times \to b ∣ ∣ ∣ = ∣ ∣ \to a ∣ ∣ ∣ ∣ ∣ \to b ∣ ∣ ∣ sin θ$

$35 = \sqrt{26} \times 7 sin θ$

$sin θ = \frac{5}{\sqrt{26}}$

We know that

$sin θ = \frac{H e i g h t}{H y p o t e n u s e}$

Therefore,

$H e i g h t = 5$

$H y p o t e n u s e = \sqrt{26}$

$B a s e = \sqrt{{(H y p o t e n u s e)}^{2} - {(H e i g h t)}^{2}}$

$B a s e = \sqrt{26 - 1}$

$B a s e = \sqrt{25} = 5$

Therefore,

$cos θ = \frac{B a s e}{H y p o t e n u s e}$

$cos θ = \frac{1}{\sqrt{26}}$

Since,

$\to a \cdot \to b = ∣ ∣ \to a ∣ ∣ ∣ ∣ ∣ \to b ∣ ∣ ∣ cos θ$

$\to a \cdot \to b = \sqrt{26} \times 7 \times \frac{1}{\sqrt{26}}$

$\to a \cdot \to b = 7$

Hence, this is the answer.

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