Define a-b- and prove that - a-b- - a- -b- tan- where - is the angle between a- and b-

a⃗×b⃗ is defined as a c⃗ that is perpendicular (orthogonal) to both a⃗ and b⃗ , with a direction given by the right hand rule and a magni ...

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Byju's Answer

Standard XII

Mathematics

Addition of Vectors

Define a⃗×b...

Question

Define $\to a \times \to b$ and prove that $∣ ∣ \to a \times \to b ∣ ∣ = (\to a . \to b) tan θ$ , where $θ$ is the angle between $\to a$ and $\to b$ .

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Solution

$\to a \times \to b$ is defined as a $\to c$ that is perpendicular (orthogonal) to both $\to a$ and $\to b$ , with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.
$L . H . S : | \to a \times \to b | = | \to a | | \to b | sin θ$
$R . H . S : (\to a . \to b) tan θ = | \to a | | \to b | cos θ tan θ = | \to a | | \to b | sin θ = L . H . S$
Hence proved.

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Define →a×→b and prove that ∣∣→a×→b∣∣=(→a.→b)tanθ, where θ is the angle between →a and →b.

Define $\to a \times \to b$ and prove that $∣ ∣ \to a \times \to b ∣ ∣ = (\to a . \to b) tan θ$ , where $θ$ is the angle between $\to a$ and $\to b$ .