We are given a rectangular-based pyramid $PQRST$, where the base $PQRS$ is a square with side lengths of 22 cm, and the height of the pyramid from the apex $T$ to the center of the base is 20 cm. We are tasked with finding the angle between the line $ST$ and the plane $PQRS$.

Approach:

1. Coordinates Setup:

   • Place the center of the square base $O$ at the origin $(0, 0, 0)$.
   • Let the vertices $P, Q, R, S$ of the square base be at:
     • $P(-11, -11, 0)$
     • $Q(11, -11, 0)$
     • $R(11, 11, 0)$
     • $S(-11, 11, 0)$
   • The apex $T$ is directly above $O$ with coordinates $(0, 0, 20)$.

2. Vector for Line $ST$: The vector $\overrightarrow{ST}$ can be calculated using the coordinates of $S$ and $T$. $\overrightarrow{ST} = (0 - (-11), 0 - 11, 20 - 0) = (11, -11, 20)$

3. Normal to the Plane $PQRS$: Since $PQRS$ is in the $xy$-plane, the normal vector to the plane is simply in the $z$-direction, which is: $\mathbf{n} = (0, 0, 1)$

4. Angle Between Line and Plane: The angle $\theta$ between the line and the plane can be found using the dot product formula between the line's direction vector $\mathbf{v} = \overrightarrow{ST}$ and the normal vector $\mathbf{n}$: $\cos \theta = \frac{|\mathbf{v} \cdot \mathbf{n}|}{|\mathbf{v}| |\mathbf{n}|}$ where:

   • $\mathbf{v} = (11, -11, 20)$
   • $\mathbf{n} = (0, 0, 1)$

   The dot product is: $\mathbf{v} \cdot \mathbf{n} = (11)(0) + (-11)(0) + (20)(1) = 20$

   The magnitude of $\mathbf{v}$ is: $|\mathbf{v}| = \sqrt{11^2 + (-11)^2 + 20^2} = \sqrt{121 + 121 + 400} = \sqrt{642} \approx 25.33$

   The magnitude of $\mathbf{n}$ is: $|\mathbf{n}| = \sqrt{0^2 + 0^2 + 1^2} = 1$

   So, we have: $\cos \theta = \frac{|20|}{25.33 \times 1} \approx \frac{20}{25.33} \approx 0.7895$

   Therefore, the angle $\theta$ is: $\theta = \cos^{-1}(0.7895) \approx 37.8^\circ$

Final Answer:

The angle between the line $ST$ and the plane $PQRS$ is approximately $37.8^\circ$.

Would you like more details on any of the steps?

Relative Questions:

1. How do you find the direction vector of a line given two points?
2. What is the geometric interpretation of the dot product in vector math?
3. How do you find the normal vector of a plane in 3D space?
4. What is the physical meaning of an angle between a line and a plane?
5. Can you calculate angles between two lines in space using similar methods?

Tip:

When dealing with geometric figures in 3D, assigning coordinates to points and using vector operations like dot products and cross products can simplify finding angles and distances.