Cross Product

Definition

The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both of the original vectors and whose magnitude is the area of the parallelogram formed by the two vectors.

5 Must Know Facts For Your Next Test

1. The cross product of two vectors $\vec{a}$ and $\vec{b}$ is denoted as $\vec{a} \times \vec{b}$.
2. The cross product is defined as $\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)$.
3. The 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 the two vectors.
4. The cross product is anticommutative, meaning that $\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$.
5. The cross product is used to find the area of a parallelogram formed by two vectors and to determine the direction of a vector that is perpendicular to two other vectors.

Review Questions

• Explain how the cross product can be used to find the area of a parallelogram formed by two vectors.
  • The cross product of two vectors $\vec{a}$ and $\vec{b}$ gives a vector that is perpendicular to both $\vec{a}$ and $\vec{b}$. The magnitude of this vector is equal to the area of the parallelogram formed by the two vectors. Specifically, the area of the parallelogram is given by $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$, where $\theta$ is the angle between the two vectors. This property of the cross product is useful in solving problems related to the area of parallelograms and triangles in the context of proportions and their applications.
• Describe how the cross product can be used to determine the direction of a vector that is perpendicular to two other vectors.
  • The cross product of two vectors $\vec{a}$ and $\vec{b}$ results in a vector $\vec{c} = \vec{a} \times \vec{b}$ that is perpendicular to both $\vec{a}$ and $\vec{b}$. This property of the cross product can be used to find the direction of a vector that is perpendicular to two given vectors. The direction of the resulting vector $\vec{c}$ is determined by the right-hand rule, which states that if the fingers of the right hand are curled in the direction of the first vector $\vec{a}$ and the second vector $\vec{b}$, then the thumb will point in the direction of the cross product $\vec{c}$. This is a useful technique in solving problems related to proportions and their applications that involve finding the direction of a vector perpendicular to two other vectors.
• Analyze how the cross product can be used to solve problems related to proportions and their applications, such as finding the missing side of a parallelogram or triangle.
  • The cross product can be used to solve various problems related to proportions and their applications, particularly those involving the area of parallelograms and triangles. For example, if the lengths of two adjacent sides of a parallelogram and the angle between them are known, the cross product can be used to find the area of the parallelogram, which is equal to the magnitude of the cross product of the two vectors representing the sides. This information can then be used to solve for a missing side length or other unknown quantity in the context of proportions and their applications. Additionally, the cross product can be used to find the direction of a vector that is perpendicular to two given vectors, which may be necessary in solving problems involving proportions and their practical applications.

Related terms

Dot Product: The dot product, also known as the scalar product, is a binary operation on two vectors that results in a scalar quantity. It is used to determine the magnitude of a vector projection or the angle between two vectors.

Vector: A vector is a mathematical object that has both magnitude and direction. Vectors are used to represent quantities that have both size and direction, such as force, velocity, and displacement.

Parallelogram: A parallelogram is a quadrilateral with two pairs of parallel sides. The area of a parallelogram is equal to the product of the lengths of its two adjacent sides and the sine of the angle between them.