The sine of the angle between the vectors $\hat{i}+3\hat{j}+2\hat{k}$ and $2\hat{i}-4\hat{j}+\hat{k}$ is
- $\sqrt{\dfrac{155}{156}}$
- $\sqrt{\dfrac{115}{116}}$
- $\sqrt{\dfrac{115}{147}}$
- $\sqrt{\dfrac{157}{158}}$
Let u = (1, 3, 2) and v = (2, -4, 1). The cross product u x v = (3*1 - 2*-4, 2*2 - 1*1, 1*-4 - 3*2) = (11, 3, -10). Magnitude |u x v| = sqrt(121 + 9 + 100) = sqrt(230). Magnitudes |u| = sqrt(1+9+4) = sqrt(14), |v| = sqrt(4+16+1) = sqrt(21). sin(theta) = |u x v| / (|u||v|) = sqrt(230) / sqrt(14*21) = sqrt(230/294) = sqrt(115/147).
The sine of the angle between two vectors is found using the cross product formula, sin theta = magnitude of A cross B divided by (magnitude of A times magnitude of B). For vectors (1, 3, 2) and (2, -4, 1), the cross product is (11, 3, -10), which has a magnitude of square root of (121 + 9 + 100) = square root of 230. The magnitudes of the vectors are square root of 14 and square root of 21, so the denominator is square root of 294. Thus, sin theta = square root of 230 / square root of 294 = square root of (115/147).