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Standard XII

Mathematics

Sign of Trigonometric Ratios in Different Quadrants

If θ is the a...

Question

If θ is the angle between any two vectors $\vec{a} and \vec{b}$ , then $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$ when θ is equal to

(a) 0
(b) π/4
(c) π/2
(d) π

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Solution

(b) π/4

$Let θ be the angle between \vec{a} and \vec{b} . We know |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin θ \vec{a} . \vec{b} = |\vec{a}| |\vec{b}| \cos θ \Rightarrow |\vec{a} . \vec{b}| = ||\vec{a}| |\vec{b}| \cos θ| = |\overset{' \to}{a}| |\vec{b}| |\cos θ| Given: |\vec{a} . \vec{b}| = |\vec{a} \times \vec{b}| \Rightarrow |\vec{a}| |\vec{b}| |\cos θ| = |\vec{a}| |\vec{b}| \sin θ \Rightarrow |\cos θ| = \sin θ \Rightarrow θ = \frac{π}{4}$

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If θ is the angle between any two vectors a →and b →, then a →· b →= a →×b → when θ is equal to (a) 0 (b) π/4 (c) π/2 (d) π

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If θ is the angle between any two vectors $\vec{a} and \vec{b}$ , then $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$ when θ is equal to

(a) 0
(b) π/4
(c) π/2
(d) π