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Let's go through each of the questions step by step.

### 1. Show that the diagonals of a rhombus intersect at right angles using the dot product.
- **Concept**: The diagonals of a rhombus intersect at right angles when their dot product is zero. Given two vectors representing the diagonals, you need to calculate the dot product and show it equals zero.

Let the diagonals be represented by vectors **d1** and **d2**. If \(\mathbf{d1} \cdot \mathbf{d2} = 0\), then the diagonals are perpendicular, confirming that they intersect at right angles.

### 2. Consider the vectors 
\[
\mathbf{u} = \begin{pmatrix} 1 \\ 1 \\ 4 \end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}
\]
#### a) Find \(\mathbf{u} \times \mathbf{v}\).
- The cross product of two vectors is computed as:
\[
\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 1 & 4 \\ 2 & -1 & 2 \end{vmatrix}
\]

#### b) Find a point-normal and Cartesian equation of the plane passing through the point \(D(0, 1, 0)\) and parallel to vectors \(\mathbf{u}\) and \(\mathbf{v}\).
- The cross product \(\mathbf{u} \times \mathbf{v}\) gives the normal vector to the plane. The point-normal equation of the plane is then:
\[
\mathbf{n} \cdot (\mathbf{r} - \mathbf{r_0}) = 0
\]
where \(\mathbf{r_0}\) is the given point \(D(0, 1, 0)\).

### 3. Find the point on the plane \(P : x + 2y - 2z = 5\) that is closest to the point \(A(5, -4, 5)\).
- The shortest distance from a point to a plane is along the line perpendicular to the plane. The formula for the distance \(d\) from a point \(A(x_1, y_1, z_1)\) to the plane \(Ax + By + Cz + D = 0\) is:
\[
d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}}
\]
Use this to calculate the distance and find the coordinates of the closest point.

### 4. Find a Cartesian, point-normal, and parametric vector equation of the plane passing through the point \((1, 2, 0)\) and containing the line:
\[
\frac{x - 5}{2} = \frac{y - 1}{2} = \frac{z - 1}{3}
\]
- The direction ratios of the line are \(\mathbf{d} = \begin{pmatrix} 2 \\ 2 \\ 3 \end{pmatrix}\).
- The plane equation must contain the point \((1, 2, 0)\) and have a normal vector perpendicular to \(\mathbf{d}\).

Would you like me to solve any of these problems in detail or provide more specific steps for each? Let me know if you want to focus on any question in particular!

Here are some related questions:
1. How is the dot product related to the angle between two vectors?
2. What is the geometrical interpretation of the cross product of two vectors?
3. How do you find the equation of a plane given a point and a normal vector?
4. How can you calculate the distance between a point and a plane?
5. What is the parametric equation of a line, and how is it used to define planes?

**Tip**: Understanding vector operations like dot product and cross product is crucial for solving geometric problems involving planes and angles.

Use the dot product to show that the diagonals of a rhombus intersect at right angles. Consider the vectors u = (1, 1, 4) and v = (2, -1, 2). Find u × v. Find the point-normal and Cartesian equation of the plane passing through D(0, 1, 0) parallel to vectors u and v. Find the point on the plane x + 2y - 2z = 5 closest to point A(5, -4, 5). Find the equation of the plane passing through (1, 2, 0) and containing the line with parametric equation x-5/2 = y-1/2 = z-1/3.

Explore a detailed solution to multiple vector algebra problems, including finding cross products, determining equations of planes, and proving the intersection of rhombus diagonals using dot products. Suitable for undergraduate-level students with a focus on vector operations and geometry.

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