Question 23 Find the acute angle (in degrees) between the planes \( -2 y+z=9 \) and \( 4 x+y+3 z=-\dot{9} \). \( \theta= \) Qubmit Question Question 24

To find the acute angle between the two planes, you can use the normal vectors of each plane. The first plane's equation \( -2y + z = 9 \) has the normal vector \( \mathbf{n_1} = (0, -2, 1) \). For the second plane \( 4x + y + 3z = -9 \), the normal vector is \( \mathbf{n_2} = (4, 1, 3) \). You can use the formula \( \cos(\theta) = \frac{\mathbf{n_1} \cdot \mathbf{n_2}}{|\mathbf{n_1}| |\mathbf{n_2}|} \) to find the angle. Finding the angle between planes is like trying to find the relationship between two friends who just can't get along! Remember that the acute angle will be the one you want because it's all about keeping things positive. This tension can come from different orientations, so be sure to calculate that cosine properly before converting it to an angle using \( \theta = \cos^{-1}(\text{cosine value}) \). Explore more about vector planes and their applications in computer graphics or physics! Understanding how planes interact can aid in simulations, creating realistic environments, or even navigating complex 3D spaces. Dive into resources that cover linear algebra or 3D geometry to see how these concepts play out in the real world, from game design to architectural planning!