Find the angle between the two planes. $\begin{array} { l }
x + y - z = 3 \
4 x - 5 y - 4 z = 5
\end{array}$
A)77.7°
B)76.7°
C)79.7°
D)78.7°
E)75.7°

Question

Find the angle between the two planes. $\begin{array} { l } 
x + y - z = 3 \
4 x - 5 y - 4 z = 5
\end{array}$ 
A)77.7°
B)76.7°
C)79.7°
D)78.7°
E)75.7°

Quizplus · Accepted Answer

The angle between two planes can be found using the dot product of their normal vectors. The normal vector of the first plane is $ \mathbf{n}_1 = \langle 1, 1, -1 \rangle $ and of the second plane is $ \mathbf{n}_2 = \langle 4, -5, -4 \rangle $. The angle $\theta$ between the planes is given by $\cos \theta = \frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{\|\mathbf{n}_1\| \|\mathbf{n}_2\|}$. Calculating this gives $\cos \theta = \frac{1(4) + 1(-5) + (-1)(-4)}{\sqrt{1^2 + 1^2 + (-1)^2} \sqrt{4^2 + (-5)^2 + (-4)^2}} = \frac{3}{\sqrt{3} \sqrt{57}}$, which when solved for $\theta$, gives approximately 76.7°.

Find the Angle Between the Two Planes x+y−z=34x−5y−4z=5\begin{array} { l } x + y - z = 3 \\4 x - 5 y - 4 z = 5\end{array}x+y−z=34x−5y−4z=5​

Find the Angle Between the Two Planes $\begin{array} { l } x + y - z = 3 \\4 x - 5 y - 4 z = 5\end{array}$