If the vectors \[\vec{a}\] and \[\vec{b}\] are such that \[\left| \vec{a} \right| = 3, \left| \vec{b} \right| = \frac{2}{3}\] and \[\vec{a} \times \vec{b}\] is a unit vector, then write the angle between \[\vec{a}\] and \[\vec{b}\]

Let the angle between \[\vec{a}\] and \[\vec{b}\] be \[\theta\] It is given that \[\vec{a} \times \vec{b}\] is a unit vector. \[\therefore \left| \vec{a} \times \vec{b} \right| = 1\]\[ \Rightarrow \left| \vec{a} \right|\left| \vec{b} \right|\sin\theta = 1\]\[ \Rightarrow 3 \times \frac{2}{3} \times \sin\theta = 1\]\[ \Rightarrow \sin\theta = \frac{1}{2}\]\[ \Rightarrow \theta = \frac{\pi}{6}\] Thus, the angle between \[\vec{a}\] and \[\vec{b}\] is \[\frac{\pi}{6}\]

If the Vectors → a and → B Are Such that | → a | = 3 , ∣ ∣ → B ∣ ∣ = 2 3 and → a × → B is a Unit Vector, Then Write the Angle Between → a and → B - Mathematics

Question

If the vectors $\vec{a}$ and $\vec{b}$ are such that $| \vec{a} | = 3, | \vec{b} | = \frac{2}{3}$ and $\vec{a} \times \vec{b}$ is a unit vector, then write the angle between $\vec{a}$ and $\vec{b}$

Solution

Let the angle between $\vec{a}$ and $\vec{b}$ be $θ$ It is given that $\vec{a} \times \vec{b}$ is a unit vector.

$∴ | \vec{a} \times \vec{b} | = 1$
$\Rightarrow | \vec{a} | | \vec{b} | \sin θ = 1$
$\Rightarrow 3 \times \frac{2}{3} \times \sin θ = 1$
$\Rightarrow \sin θ = \frac{1}{2}$
$\Rightarrow θ = \frac{π}{6}$

Thus, the angle between $\vec{a}$ and $\vec{b}$ is $\frac{π}{6}$

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Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors

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Chapter 24 Scalar Or Dot Product
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