Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian.
-8x + 4y + 3z = 1 and 7x + 10y + 7z = -6
A) 1.079 rad
B) 0.492 rad
C) 1.534 rad
D) 1.458 rad

Question

Quizplus · Accepted Answer

To find the acute angle between the planes, we need to find the dot product of their normal vectors and then use the formula $	heta = \cos^{-1}\left(\frac{\vec{n_1}\cdot\vec{n_2}}{\left|\vec{n_1}ight|\left|\vec{n_2}ight|}ight)$, where $\vec{n_1}$ and $\vec{n_2}$ are the normal vectors of the planes.

The normal vectors of the planes can be found by looking at the coefficients of the variables. For the first plane, the normal vector is $\vec{n_1} = \begin{pmatrix}8 \ 4 \ 3\end{pmatrix}$, and for the second plane, the normal vector is $\vec{n_2} = \begin{pmatrix}7 \ 10 \ 7\end{pmatrix}$.

The dot product of $\vec{n_1}$ and $\vec{n_2}$ is $8\cdot7+4\cdot10+3\cdot7=113$, and the magnitudes of $\vec{n_1}$ and $\vec{n_2}$ are $\left|\vec{n_1}ight|=\sqrt{8^2+4^2+3^2}=9$, and $\left|\vec{n_2}ight|=\sqrt{7^2+10^2+7^2}=15$.

Therefore, $	heta = \cos^{-1}\left(\frac{113}{9\cdot15}ight) \approx 0.492$ radians (rounded to 3 decimal places).

Thus, the best choice is B.

Use a Calculator to Find the Acute Angle Between the Planes