Question

For direction ratios \[(a_1,b_1,c_1)\] and \[(a_2,b_2,c_2)\], which expression gives \[\sin\theta\]?

Options

• \[\sin\theta=\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}+\sqrt{a_2^2+b_2^2+c_2^2}}\]

• \[\sin\theta=\frac{\sqrt{(a_1b_2-a_2b_1)^2+(b_1c_2-b_2c_1)^2+(c_1a_2-c_2a_1)^2}}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\]

• \[\sin\theta=\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}\]

• \[\sin\theta=\left|\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\right|\]

Answer 1

The numerator contains the three cross-product component expressions. It is divided by the product of the magnitudes of the direction-ratio triples.