If and , then what is the value of ?

3. By Lagrange's identity, . Substituting the given values, . This gives . Since magnitude is non-negative, .

If (\vec{a}\times\vec{b})^{2}+(\vec{a}\cdot\vec{b})^{2}=144 and |\vec{b}|=4, then what is the value of |\vec{a}|?

A. 3 ✓
B. 4
C. 5
D. 6

Correct Answer: A. 3

Explanation

By Lagrange's identity, |\vec{a}\times\vec{b}|^2 + (\vec{a}\cdot\vec{b})^2 = |\vec{a}|^2|\vec{b}|^2. Substituting the given values, 144 = |\vec{a}|^2(4)^2. This gives 16|\vec{a}|^2 = 144 \implies |\vec{a}|^2 = 9. Since magnitude is non-negative, |\vec{a}| = 3.

If (\vec{a}\times\vec{b})^{2}+(\vec{a}\cdot\vec{b})^{2}=144 and |\vec{b}|=4, then what is the value of |\vec{a}|?

Explanation

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