Enrichment, Extension, and Application

1 point

Exercises

1. Consider a point $$P$$ and a plane $$\pi$$ with equation $$3x - 2y + 2z = 0$$. Let $$Q$$ be the point of intersection of the plane $$\pi$$ and the line through $$P$$ with direction vector $$(1, 0, -2)$$, and let $$R$$ be the point of intersection of the plane $$\pi$$ and the line through $$P$$ with direction vector $$(1, 1, 0)$$.
1. If the point $$P$$ is on the line with equation $$x = y = z$$, find all possible positions of $$P$$ such that the length of $$QR$$ is $$27$$.
2. If the point $$P$$ is on the line with equation $$x = 0$$, find all possible positions of $$P$$ such that the length of $$QR$$ is $$24$$.
1. Prove that the scalar equation of the plane that cuts the axes at $$A~(a, 0, 0)$$, $$B~(0, b, 0)$$, and $$C~(0, 0, c)$$ is given by $$\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{z}{c} = 1$$.
2. Calculate the value of the solid bounded by the planes with equations $$3x + 2y + 5z = 120$$, $$5x + 4y + 6z = 120$$, $$x = 0$$, $$y = 0$$, and $$z = 0$$.
2. The cube $$ABCDEFGH$$ with an edge length of $$2$$ is sliced with a plane passing through the midpoints of edge $$AE$$, $$BC$$, and $$GH$$, as shown.

Find the shape of the section in the cube and calculate its area.

3. Given the three lines with parametric equations $$L_1: \begin{cases} x = 1 \\ y = 2 - 2a \\ z = a + 3 \end{cases}, L_2: \begin{cases} x = 17 - 8b \\ y = 6 - 2b \\ z = 9 - 3b \end{cases}, L_3: \begin{cases} x = 1 \\ y = 6c - 4 \\ z = 3c \end{cases}$$, where $$a, b, c$$ are real numbers, and the plane with equation $$x + 2y - 2z + 7 = 0$$:
1. Prove that the three lines are concurrent, and find the coordinates of the intersection point, $$V$$.
2. Find the intersection points of the three lines with the plane, denoted $$A, B, C$$ respectively for $$L_1, L_2, L_3$$.
3. Find the volume of tetrahedron $$VABC$$.
1. Show that the set of equations $$\begin{cases} x = 6 - p - t \\ y = p \\ z = t \end{cases}$$, $$0 \leq p \leq 2$$, $$0 \leq t \leq 3$$ bounds a region whose interior is in the shape of a parallelogram.
2. A parallelepiped is bordered by the pairs of parallel planes: \begin{align*} P_1: x + 2y + 2z &= 3 & P_2: x + 2y + 2z &= 6 \\ P_3: x - 2y - 2z &= 7 & P_4: x - 2y - 2z &= 19 \\ P_5: x + 2y - 2z &= 11 & P_6: x + 2y - 2z &= 19 \\ \end{align*}
1. Find the distance between the planes $$P_1$$ and $$P_2$$.
2. Find the point $$A$$ where $$P_1$$, $$P_3$$, and $$P_5$$ meet.
3. Find the volume of the parallelepiped.
1. Explain why $$x^2 + y^2 + z^2 = 50$$ is the equation of a sphere centred at the origin.
2. Find the center and the radius of the sphere with equation $$(x - 3)^2 + (y - 5)^2 + (z + 4)^2 = 289$$.
3. Explain why $$x^2 + y^2 - z^2 = 0$$ is the equation of a cone.
4. Find the intersection between the sphere in part a) and the cone given in part b). Give both the equation and the interpretation of this figure.
5. How many points $$P~(a, b, c)$$ with $$a, b, c \in \mathbb{Z}$$ are situated on the figure found in part d)?
4. The intersection between the sphere with equation $$(x - 3)^2 + (y - 5)^2 + (z + 4)^2 = 289$$ and the plane with equation $$6x + 2y - 9z = k$$ is a circle of radius $$15$$.
1. Find both possible values for $$k$$.
2. For each value of $$k$$ found in part a), find the coordinates of the centre of the circle.
5. The Cartesian equation of a sphere centred at the origin $$O$$ is $$x^2 + y^2 + z^2 = R^2$$. A set of parametric equations for the same sphere are given by $\begin{cases} x = R\cos{( \theta )}\sin{( \phi )} \\ y = R\sin{( \theta )}\sin{( \phi )} \\ z = R\cos{( \phi )} \end{cases}$ where $$\theta \in [0, 360^\circ)$$ is the angle that the projection of the line $$OP$$ onto the $$xy$$-plane makes with the positive $$x$$-axis ($$\theta = 0^\circ$$ corresponds to $$OP$$ lying coincident with the positive $$x$$-axis, $$\theta = 180^\circ$$ corresponds to $$OP$$ lying coincident with the negative $$x$$-axis), and $$\phi \in [0, 180^\circ]$$ is the angle that the line $$OP$$ makes with the positive $$z$$-axis ($$\phi = 0^\circ$$ corresponds to $$OP$$ lying coincident to the positive $$z$$-axis, $$\phi = 180^\circ$$ corresponds to $$OP$$ lying coincident to the negative $$z$$-axis). These equations are also called the spherical equations; $$R$$, $$\theta$$, and $$\phi$$ are called the spherical coordinates of the point $$P$$ on the sphere.
1. Show that any point $$P~\Big( R\cos{(\theta)}\sin{(\phi)}, R\sin{(\theta)}\sin{(\phi)}, R\cos{(\phi)} \Big)$$ satisfies the Cartesian equation $$x^2 + y^2 + z^2 = R^2$$.
2. Find a point $$P$$ that has positive integers as coordinates and is situated on the sphere of Cartesian equation $$x^2 + y^2 + z^2 = 49$$. Find the polar coordinates of this point. (Round the angles to the closest degree.)
6. Consider the following system $\begin{cases} x^2 + y^2 + z^2 = 6 \\ z = x^2 + y^2 - 4 \\ x^2 = (y + z)^2 \end{cases}, x, y, z \in \mathbb{R}$
1. Give a geometric interpretation of each equation in the system.
2. Solve the system and interpret the solution.
7. The intersection between the sphere $$x^2 + y^2 + z^2 = 169$$ and the plane $$2x + 3y + 6z = 35$$ is a circle, $$\mathcal{C}$$. Find the centres of the spheres of radius $$15$$ that contain the circle $$\mathcal{C}$$.