Question

Let \(P, Q, R,\) and \(S\) be four points, \(n o t\) all on one plane, as in the diagram. Show that the volume of the pyramid they determine is $$\frac{1}{6}|\overrightarrow{P Q} \cdot(\overrightarrow{P R} \times \overrightarrow{P S})|$$

Short answer

The volume is \( \frac{1}{6}|\overrightarrow{PQ} \cdot (\overrightarrow{PR} \times \overrightarrow{PS})| \)."

Step-by-step solution

Understanding the Position Vectors

To compute the volume of the pyramid formed by points \( P, Q, R, \) and \( S \), we begin by considering \( P \) as a reference point in 3D space. The position vectors relative to \( P \) are \( \overrightarrow{PQ} \), \( \overrightarrow{PR} \), and \( \overrightarrow{PS} \). This allows us to express these vectors as extensions in space from the base point \( P \).

Finding the Cross Product

The next step is to compute the cross product \( \overrightarrow{PR} \times \overrightarrow{PS} \). This cross product yields a vector that is orthogonal to both \( \overrightarrow{PR} \) and \( \overrightarrow{PS} \), meaning it is perpendicular to the plane formed by the triangle \( PRS \).

Calculating the Scalar Triple Product

Now, compute the dot product \( \overrightarrow{PQ} \cdot (\overrightarrow{PR} \times \overrightarrow{PS}) \). This dot product is known as the scalar triple product, which provides a scalar value. The absolute value of this scalar gives the measure of the parallelepiped's volume formed by these vectors.

Finding the Volume of the Pyramid

The volume of the pyramid is one-sixth the volume of the parallelepiped, as a pyramid is essentially a three-sided base with a height extending to the last point forming a fourth vertex. Thus, the volume of the pyramid is \( \frac{1}{6} \) of the absolute value of the scalar triple product, \( \frac{1}{6}|\overrightarrow{PQ} \cdot (\overrightarrow{PR} \times \overrightarrow{PS})| \).

Key concepts

Cross Product

The cross product is a mathematical operation used in vector calculus to find a new vector that is orthogonal, or perpendicular, to two input vectors. This can be very useful in 3D geometry because it helps define planes and surfaces in space.

When you perform a cross product on two vectors, say \(\overrightarrow{PR}\) and \(\overrightarrow{PS}\), you perform a series of calculations that include the components of these vectors. The resulting vector from \(\overrightarrow{PR} \times \overrightarrow{PS}\) will point in a direction that is perpendicular to the plane containing \(PR\) and \(PS\).

• The magnitude of the cross product gives the area of the parallelogram formed by the two vectors.
• The direction is determined by the right-hand rule, which involves pointing the index finger in the direction of the first vector and the middle finger in the direction of the second vector. Your thumb then points in the direction of the cross product.

This property is essential for understanding the geometrical interpretations of vectors in space, particularly when dealing with volumes and areas.

Scalar Triple Product

The scalar triple product is a combination of the dot product and the cross product. It involves three vectors and results in a scalar quantity (a real number).

To compute a scalar triple product, we first find the cross product of two vectors (e.g., \(\overrightarrow{PR} \times \overrightarrow{PS}\)) and then take the dot product of this resulting vector with a third vector (\(\overrightarrow{PQ}\)). This is mathematically represented by \(\overrightarrow{PQ} \cdot (\overrightarrow{PR} \times \overrightarrow{PS})\).

• This result represents the volume of a parallelepiped in 3D space.
• A positive result indicates the vectors form a right-handed system, whereas a negative result indicates a left-handed system.
• The absolute value of this result gives the exact volume measurement regardless of direction.

This scalar value is crucial for various applications such as determining volumes and solving physics problems using vector analysis.

Volume of a Pyramid

Calculating the volume of a geometric shape like a pyramid can be exciting because it links algebra with spatial visualizations.

For the pyramid formed by the points \(P, Q, R,\) and \(S\), its volume can be derived from the scalar triple product previously calculated. The relationship involves a simple factor: the volume of a pyramid is one-sixth that of a parallelepiped formed by the same base and height vectors. Therefore, the volume can be expressed as:

\[\text{Volume of Pyramid} = \frac{1}{6}|\overrightarrow{PQ} \cdot (\overrightarrow{PR} \times \overrightarrow{PS})|\]

• The factor of \(\frac{1}{6}\) accounts for the fact that a pyramid is essentially one third of a prism that could be built using the base as one face.
• This calculation assumes a correct orientation of the vectors, often accompanied by a positive scalar triple product outcome.

Hence, through these vector operations, one can precisely determine the volume of complex 3D shapes like pyramids.