Problem 11 - Advanced Engineering Mathematics

Question

find \(\overrightarrow{P_{1} P_{2}} \times \overrightarrow{P_{1} P_{3}}\) $$ P_{1}(2,1,3), P_{2}(0,3,-1), P_{3}(-1,2,4) $$

Step-by-Step Solution

Answer

The cross product is \((6, 14, 4)\).

Step 1: Find the Vector \(\overrightarrow{P_{1}P_{2}}\)

Use the coordinates of points \(P_1\) and \(P_2\) to find the vector. The formula is \(\overrightarrow{P_{1}P_{2}} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)\). Thus, \(\overrightarrow{P_{1}P_{2}} = (0 - 2, 3 - 1, -1 - 3) = (-2, 2, -4)\).

Step 2: Find the Vector \(\overrightarrow{P_{1}P_{3}}\)

Use the coordinates of points \(P_1\) and \(P_3\) to find the vector. The formula is \(\overrightarrow{P_{1}P_{3}} = (x_3 - x_1, y_3 - y_1, z_3 - z_1)\). Therefore, \(\overrightarrow{P_{1}P_{3}} = (-1 - 2, 2 - 1, 4 - 3) = (-3, 1, 1)\).

Step 3: Calculate the Cross Product

To find \(\overrightarrow{P_{1}P_{2}} \times \overrightarrow{P_{1}P_{3}}\), use the cross product formula for vectors \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\), which is \((b_1c_2 - c_1b_2, c_1a_2 - a_1c_2, a_1b_2 - b_1a_2)\). Applying this: \((2 \cdot 1 - (-4) \cdot 1, -4 \cdot (-3) - (-2) \cdot 1, -2 \cdot 1 - 2 \cdot (-3)) = (2 + 4, 12 + 2, -2 + 6) = (6, 14, 4)\).

Step 4: Present the Result

The cross product vector \(\overrightarrow{P_{1}P_{2}} \times \overrightarrow{P_{1}P_{3}}\) is \((6, 14, 4)\).

Key Concepts

Cross Product

The cross product, also known as the vector product, is a fundamental operation in vector calculus. This operation is used to calculate a vector that is perpendicular to two given vectors in three-dimensional space. It is especially useful in physics and engineering for determining torque, angular momentum, and in various applications involving force and motion.

The formula for finding the cross product of two vectors, \(\vec{A} = (a_1, b_1, c_1)\) and \(\vec{B} = (a_2, b_2, c_2)\), is:

• \(\vec{A} \times \vec{B} = (b_1c_2 - c_1b_2, c_1a_2 - a_1c_2, a_1b_2 - b_1a_2)\)

This operation involves a specific pattern of subtraction and multiplication that results in a new vector oriented perpendicular to the original two.
Understanding the cross product helps solve problems involving levels of perpendicular detail, such as calculating the area of a parallelogram defined by two vectors or determining the normal vector to a plane.

Vector Operations

Vector operations form the building blocks of vector calculus, involving various operations that enable manipulation and analysis of vectors. The basic operations include addition, subtraction, scalar multiplication, dot product, and cross product.

Here's a brief guide on these operations:

• Addition and Subtraction: Vectors are added or subtracted by combining their corresponding components. For example, to add vectors \(\vec{A} = (a_1, b_1, c_1)\) and \(\vec{B} = (a_2, b_2, c_2)\), you perform the operation \(\vec{A} + \vec{B} = (a_1 + a_2, b_1 + b_2, c_1 + c_2)\).
• Scalar Multiplication: Multiply each component of the vector by the scalar. For vector \(\vec{A} = (a_1, b_1, c_1)\) and a scalar \(k\), the result is \(k\vec{A} = (ka_1, kb_1, kc_1)\).
• Dot Product: This operation gives a scalar result and is defined as \(\vec{A} \cdot \vec{B} = a_1a_2 + b_1b_2 + c_1c_2\).
• Cross Product: Produces a vector, essential in finding orthogonal vectors, described in detail in the cross product section.

Master these operations to harness the full potential of vector analysis in various multidimensional spaces.

Coordinate Geometry

Coordinate geometry merges algebra and geometry to describe and explore geometric figures using a coordinate system. It allows the translation of geometric conditions into algebraic equations and vice versa.

Here are some key terms and operations within coordinate geometry:

• Points in Space: Represented as ordered pairs or triples \( (x, y, z) \) in 3D space, defining positions within the coordinate system.
• Vector Representation: Vectors describe the movement from one point to another using coordinates, such as \(\overrightarrow{AB} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)\).
• Distance Formula: Calculating the distance between two points is crucial in analyzing geometric shapes, given by \((d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2})\).
• Plane and Line Equations: Planes in 3D coordinate systems can be described using equations like \(Ax + By + Cz = D\), which also relates to normal vectors created using cross products.

The application of coordinate geometry is vast, spanning from spatial understanding to complex geometrical problem-solving.