14.3 Cross Product and Applications

The cross product is a powerful tool for working with vectors in 3D space. It creates a new vector perpendicular to two given vectors, with applications in physics and geometry. This operation helps us calculate torque, find areas of parallelograms, and determine volumes of parallelepipeds.

Cross products build on our understanding of vector operations, expanding our toolkit for solving problems in three dimensions. By combining cross products with dot products, we can perform complex calculations and gain deeper insights into spatial relationships between vectors.

Definition and Properties of Cross Product

Calculating the Cross Product

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• Cross product (also known as vector product) operates on two vectors in three-dimensional space and produces a vector that is perpendicular to both of the vectors being multiplied
• Denoted as $\vec{a} \times \vec{b}$, where $\vec{a}$ and $\vec{b}$ are the two vectors being multiplied
• Calculated using the determinant of a matrix formed by the unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$ and the components of $\vec{a}$ and $\vec{b}$: $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} = (a_2b_3 - a_3b_2)\hat{i} - (a_1b_3 - a_3b_1)\hat{j} + (a_1b_2 - a_2b_1)\hat{k}$

Properties of the Cross Product

• Cross product is not commutative, meaning $\vec{a} \times \vec{b} \neq \vec{b} \times \vec{a}$
• Anticommutative property: $\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$
• Distributive over addition: $\vec{a} \times (\vec{b} + \vec{c}) = (\vec{a} \times \vec{b}) + (\vec{a} \times \vec{c})$
• Not associative: $(\vec{a} \times \vec{b}) \times \vec{c} \neq \vec{a} \times (\vec{b} \times \vec{c})$
• Magnitude of the cross product is given by $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$, where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$

Determining the Direction of the Cross Product

• Direction of the cross product is determined by the right-hand rule
• Point the index finger of your right hand in the direction of the first vector $\vec{a}$ and your middle finger in the direction of the second vector $\vec{b}$
• Your thumb will point in the direction of the cross product $\vec{a} \times \vec{b}$
• Example: If $\vec{a}$ points along the positive x-axis and $\vec{b}$ points along the positive y-axis, then $\vec{a} \times \vec{b}$ will point along the positive z-axis

Applications of Cross Product

Torque

• Torque is a measure of the turning force acting on an object
• Calculated using the cross product of the position vector $\vec{r}$ (from the axis of rotation to the point where the force is applied) and the force vector $\vec{F}$: $\vec{\tau} = \vec{r} \times \vec{F}$
• Magnitude of torque is given by $|\vec{\tau}| = |\vec{r}||\vec{F}|\sin\theta$, where $\theta$ is the angle between $\vec{r}$ and $\vec{F}$
• Direction of torque is perpendicular to the plane formed by $\vec{r}$ and $\vec{F}$, determined by the right-hand rule

Area of Parallelogram

• Cross product can be used to find the area of a parallelogram spanned by two vectors $\vec{a}$ and $\vec{b}$
• Area of the parallelogram is given by the magnitude of the cross product: $A = |\vec{a} \times \vec{b}|$
• This formula works because the magnitude of the cross product is equal to the base (magnitude of one vector) times the height (magnitude of the other vector times the sine of the angle between them)

Volume of Parallelepiped

• Volume of a parallelepiped (a three-dimensional figure formed by six parallelograms) can be found using the triple scalar product of three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ that form the edges of the parallelepiped
• Volume is given by the absolute value of the triple scalar product: $V = |(\vec{a} \times \vec{b}) \cdot \vec{c}|$
• Geometrically, this represents the volume of the parallelepiped because the cross product $\vec{a} \times \vec{b}$ gives a vector perpendicular to the base with a magnitude equal to the base area, and the dot product with $\vec{c}$ projects this vector onto the height of the parallelepiped

Triple Scalar Product

Definition and Properties

• Triple scalar product (also known as scalar triple product) is the dot product of a vector with the cross product of two other vectors: $(\vec{a} \times \vec{b}) \cdot \vec{c}$
• Can be calculated using the determinant of a matrix formed by the components of the three vectors: $(\vec{a} \times \vec{b}) \cdot \vec{c} = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} = a_1(b_2c_3 - b_3c_2) - a_2(b_1c_3 - b_3c_1) + a_3(b_1c_2 - b_2c_1)$
• Triple scalar product is invariant under cyclic permutations of the vectors: $(\vec{a} \times \vec{b}) \cdot \vec{c} = (\vec{b} \times \vec{c}) \cdot \vec{a} = (\vec{c} \times \vec{a}) \cdot \vec{b}$
• Changing the order of the vectors in a way that is not a cyclic permutation changes the sign of the triple scalar product: $(\vec{a} \times \vec{b}) \cdot \vec{c} = -(\vec{b} \times \vec{a}) \cdot \vec{c}$

Geometric Interpretation

• Geometrically, the triple scalar product represents the volume of the parallelepiped formed by the three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$
• If the triple scalar product is zero, the three vectors are coplanar (lie in the same plane)
• Sign of the triple scalar product depends on the orientation of the vectors:
  • Positive if the vectors form a right-handed system (when placed tail-to-tail, they follow the right-hand rule)
  • Negative if the vectors form a left-handed system