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Solve the following.
If â is unit vector and (2x⃗ - 3â) · (2x⃗ + 3â) = 91, find the value of |x⃗|.
If $\hat{a}$ is unit vector and $(2\vec{x} - 3\hat{a}) \cdot (2\vec{x} + 3\hat{a}) = 91$, find the value of $|\vec{x}|$.
Answer
Answer
$(2\vec{x} - 3\hat{a})\cdot(2\vec{x} + 3\hat{a}) = 4|\vec{x}|^2 - 9|\hat{a}|^2 = 91$
$|\hat{a}| = 1$, so $4|\vec{x}|^2 = 100$, $|\vec{x}|^2 = 25$
Hence $|\vec{x}| = 5$.
Final answer: 5