Use the cross product to determine the angle betweeen the vectors, assuming that . Round to the nearest thousandth.

Question

Use the cross product to determine the angle betweeen the vectors, assuming that  . Round to the nearest thousandth.

Quizplus · Accepted Answer

We can use the formula for the angle between two vectors, which is $	heta = \cos^{-1}\left(\frac{\mathbf{a}\cdot\mathbf{b}}{\|\mathbf{a}\|\|\mathbf{b}\|}ight)$, where $\mathbf{a}$ and $\mathbf{b}$ are the given vectors. The cross product of the vectors is $\mathbf{a}	imes\mathbf{b} = (-0.042, -0.003, -0.733)$. Since one of the components is negative, we know that the angle between the vectors is obtuse (greater than 90 degrees). Now we just need to compute the dot product: $\mathbf{a}\cdot\mathbf{b} = (11)(-1)+(4)(-4)+(2)(35) = -11-16+70 = 43$. The magnitudes of the vectors are $\|\mathbf{a}\| = \sqrt{11^2+4^2+2^2} = \sqrt{141}$ and $\|\mathbf{b}\| = \sqrt{1^2+3^2+4^2} = \sqrt{26}$. Putting it all together, we have $	heta = \cos^{-1}\left(\frac{43}{\sqrt{26}\sqrt{141}}ight) \approx 0.658$ to the nearest thousandth. Therefore, the best choice is C.

Use the Cross Product to Determine the Angle Betweeen the Vectors