Question

Show that every plane containing \(P(1,2,-1)\) and \(Q(2,0,1)\) must also contain \(R(-1,6,-5)\)

Short answer

All three points are collinear, so any plane through \(P\) and \(Q\) also contains \(R\).

Step-by-step solution

Find the Vector PQ

To determine the equation of the plane containing points \(P\), \(Q\), and any given point like \(R\), we first need to calculate the vector from \(P\) to \(Q\). The vector representation is found by subtracting the coordinates of \(P\) from \(Q\). Therefore, \(\overrightarrow{PQ} = Q - P = (2-1, 0-2, 1-(-1)) = (1, -2, 2)\).

Verify if R Lies on the Plane

We need to verify if point \(R\) lies on the plane defined by points \(P\) and \(Q\). A plane is described by a point in the plane and a normal vector perpendicular to the plane. To check if a plane can exist through all three points, find vectors \(\overrightarrow{PR}\) and \(\overrightarrow{PQ}\), then find their cross product. If the cross product is the zero vector, the points are collinear, which is not possible in this case.

Compute the Vector PR

Calculate the vector from \(P\) to \(R\): \(\overrightarrow{PR} = R - P = (-1-1, 6-2, -5-(-1)) = (-2, 4, -4)\).

Calculate Cross Product of PQ and PR

Calculate the cross product \(\overrightarrow{PQ} \times \overrightarrow{PR}\) to find a normal vector to the plane. This cross product will show the direction perpendicular to the plane containing \(P\), \(Q\), and \(R\): \[\begin{vmatrix}i & \mathbf{i} & \mathbf{j} & \mathbf{k} \ & 1 & -2 & 2 \ & -2 & 4 & -4 \end{vmatrix} = \mathbf{i}(0) - \mathbf{j}(0) + \mathbf{k}(0) = \mathbf{0}\]The result is the zero vector \(\mathbf{0}\), which confirms that the vectors are linearly dependent.

Conclusion

Since the cross product is zero, it confirms that the vectors \(\overrightarrow{PQ}\) and \(\overrightarrow{PR}\) are parallel, meaning all three points are collinear or lie in the same plane. Thus, any plane defined by \(P\) and \(Q\) will also contain \(R\).

Key concepts

Vector Operations

When dealing with plane geometry, vector operations are crucial in understanding how different points relate to each other in space. A vector is a mathematical entity that has both a magnitude and a direction. It can be used to represent the space between two points, such as point P and point Q in this exercise.

Vector operations allow us to perform calculations such as addition, subtraction, and finding magnitudes. The operation used here is subtraction to find the vector from one point to another. The formula for finding a vector \( \overrightarrow{AB} \) between two points A(x_1, y_1, z_1) and B(x_2, y_2, z_2) is given by subtracting the coordinates of A from B:

• \( \overrightarrow{AB} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \)

Understanding these operations is vital as they form the basis for more complex calculations like determining if points are collinear or lie on the same plane.

Cross Product

The cross product is a vector operation used in three-dimensional space to find a vector that is perpendicular to two other vectors. This is especially useful in geometry when determining the orientation of a plane. The cross product of two vectors \( \overrightarrow{A} \) and \( \overrightarrow{B} \) results in a third vector that is orthogonal to both \( \overrightarrow{A} \) and \( \overrightarrow{B} \).

The formula for the cross product is given by the determinant:
\[ \overrightarrow{A} \times \overrightarrow{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ \end{vmatrix} = \mathbf{i}(a_2b_3 - a_3b_2) - \mathbf{j}(a_1b_3 - a_3b_1) + \mathbf{k}(a_1b_2 - a_2b_1)\]

• This operation helps in determining the normal vector of a plane.
• If the result is the zero vector \( \mathbf{0} \), it indicates that the vectors are parallel or collinear.

In our exercise, computing the cross product of \( \overrightarrow{PQ} \) and \( \overrightarrow{PR} \) resulted in the zero vector, confirming that all three points are collinear and lie on the same plane.

Collinearity

Collinearity in geometry refers to points that lie on the same straight line. It's an important property that helps establish relationships between points in space.

When three points such as P, Q, and R are collinear, any plane that passes through any two of them will also pass through the third. In this exercise, we determine collinearity by deriving the vectors between points and calculating their cross product.

If the cross product of two vectors derived from these points results in the zero vector, it indicates that the points are indeed collinear.

• Collinearity can be tested using vector operations as shown by the cross product solution.
• If \( \overrightarrow{PQ} \times \overrightarrow{PR} = \mathbf{0} \), points P, Q, and R are collinear.

This concept is fundamental in determining whether different points lie on the same line or plane, as shown in our exercise.

Coordinates

Coordinates are numerical values that provide the position of a point in space. In plane geometry, these are often given in 3D space as (x, y, z) values. Understanding how to work with coordinates is essential for visualizing geometric concepts and solving problems.

In this exercise, the coordinates of points P(1, 2, -1), Q(2, 0, 1), and R(-1, 6, -5) are used to determine their spatial relationship. Working with coordinates involves:

• Finding vectors from one point to another by subtracting their coordinates.
• Using these vectors to determine the relative position of points.

Coordinates offer a way to handle complex geometric configurations using simple numerical methods. Mastering the art of using coordinates allows for more advanced exploration in geometry, such as defining planes and lines.