Question

a=2i-3j-k, b=i+4j+3k then what;s their angle? if i use dot product formula it comes arc cos[(-sqrt(13)/{(2*sqrt(7)}]

and if i use cross product formula then arc sin[(sqrt(15/28)] . two angles are not same if i convert to degree , then 1st one comes 132.951978120924 and the 2nd one 47.048021879076. it become equal that time if (180-47.048021879076) . because sin(180-x)=sin x . but for inverse function , we use the least value. like arc sin(1/2)= 30 degree, not 150 degree. so if i think at this angle the angle can't be equal. so which one is correct?

Answer 1

Answer on Question #51716 - Math - Vector Calculus

$a = 2i - 3j - k$, $b = i + 4j + 3k$ then what is their angle? if i use dot product formula it comes arc $\cos[\{-sqrt(13)/\{2*sqrt(7)\}]$ and if i use cross product formula then arc $\sin[\{-sqrt(15/28)\}]$. two angles are not same if i convert to degree, then 1st one comes 132.951978120924 and the 2nd one 47.048021879076. It becomes equal that time if (180-47.048021879076). because $\sin(180-x)=\sin x$. but for inverse function, we use the least value. like arc $\sin(1/2)=30$ degree, not 150 degree. so if i think at this angle the angle can't be equal. so which one is correct?

Solution:

The figure represents the unit circle. Abscissa (x-coordinate) represents the cosine of the angle theta, and the ordinate (y-coordinate) represents the sine of the angle theta. Given $\cos \theta = -\frac{\sqrt{13}}{2\sqrt{7}}$, $\sin \theta = \frac{\sqrt{15}}{2\sqrt{7}}$, the correct answer is $\theta = 132.951978120924{}^\circ$, since $\sin \theta > 0$ and $\cos \theta < 0$ (the angle $\theta$ should be greater than $90{}^\circ$, but less than $180{}^\circ$ for these values of sine and cosine).

Answer: 132.951978120924°.

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