# 3D Vectors Problem - Find Cross Product

• TheSerpent
In summary, given the length of vectors \vec{a} and \vec{b} and the angle between them, it is possible to find the length of their cross product using the formula |\vec{a} \times \vec{b}| = |\vec{a}| * |\vec{b}| * Sinθ. However, without additional information about the coordinates of the vectors, it is not possible to find the actual cross product vector.
TheSerpent

## Homework Statement

Given |$\vec{a}$| = 8, |$\vec{b}$| = 9 and the angle between vector $\vec{a}$ and $\vec{b}$ is 48° find the cross product, $\vec{a}$ X $\vec{b}$.

## Homework Equations

Let θ = angle between $\vec{a}$ and $\vec{b}$.

$\vec{a}$ . $\vec{b}$ = ( $\vec{x}$1 * $\vec{x}$2 ) + ( $\vec{y}$1 * $\vec{y}$2 ) + ( $\vec{z}$1 * $\vec{z}$2 ) = |$\vec{a}$| * |$\vec{b}$| * Cosθ

|$\vec{a}$ X $\vec{b}$| = |$\vec{a}$| * |$\vec{b}$| * Sinθ

$\vec{a}$ X $\vec{b}$ = [ ( $\vec{y}$1 * $\vec{z}$2 ) - ( $\vec{y}$2 * $\vec{z}$1 ) , ( $\vec{z}$1 * $\vec{x}$2 ) - ( $\vec{z}$2 * $\vec{x}$1 ) , ( $\vec{x}$1 * $\vec{y}$2 ) - ( $\vec{x}$2 * $\vec{y}$1 ) ]

## The Attempt at a Solution

Honestly have no idea how to work this out, the only thing I thought of was assuming the coordinates of one of the vectors. Such as $\vec{a}$ = [0,8,0]. With that use it to solve for the coordinates of $\vec{b}$ with the dot product formula then find the cross product between $\vec{a}$ and $\vec{b}$. Probably not the right way to do the question though, there might be a formula or method I am not aware.

The data given is sufficient to find the length of $\vec{a}\times\vec{b}$ but not to find $\vec{a}\times\vec{b}$. To see that just imagine rotating the vectors $\vec{a}$ and $vec{b}$ while maintaining the same lengths and angle between them. Clearly the resultant vector will shift direction as you do that.

## 1. What is a cross product in 3D vector problems?

A cross product is a mathematical operation performed on two vectors in three-dimensional space that results in a third vector that is perpendicular to both of the original vectors. It is often used in physics and engineering to calculate the torque or force exerted on an object.

## 2. How do I find the cross product of two 3D vectors?

To find the cross product of two 3D vectors, you can use the following formula:
(v₁,v₂,v₃) × (w₁,w₂,w₃) = (v₂w₃ - v₃w₂, v₃w₁ - v₁w₃, v₁w₂ - v₂w₁)
This will give you a third vector that is perpendicular to both of the original vectors.

## 3. Can the cross product of two 3D vectors be zero?

Yes, the cross product of two 3D vectors can be zero if the vectors are parallel or if one of the vectors is zero. This means that the two vectors lie on the same plane and there is no perpendicular vector between them.

## 4. What is the geometric interpretation of a cross product?

The geometric interpretation of a cross product is that the resulting vector is perpendicular to both of the original vectors. This means that the cross product can be used to find the normal vector of a plane, the direction of rotation, or the direction of a force.

## 5. How is the cross product used in real-life applications?

The cross product is used in various real-life applications, including physics, engineering, and computer graphics. It can be used to calculate the torque on an object, the magnetic field created by a current, and the direction of a force on a lever. In computer graphics, the cross product is used to create 3D effects and to determine the orientation of objects in a 3D space.

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