Why is the cross product perpendicular?

In summary, the cross product of two vectors is perpendicular to the plane that the two vectors lie on. This can be proven by showing that the dot product of the cross product with either of the two vectors is zero. The cross product is defined by the matrix definition, which allows it to produce a vector perpendicular to two given vectors. This makes it a useful tool in many applications.
  • #1
JizzaDaMan
48
0
Why is the cross product of two vectors perpendicular to the plane the two vectors lie on?

I am aware that you can prove this by showing that:

[itex](\vec{a}\times\vec{b})\cdot\vec{a} = (\vec{a}\times\vec{b})\cdot\vec{b} = 0[/itex]

Surely it was not defined as this and worked backwards though. I see little advantage in making this definition, and simply guessing it seems a bit random, so what brings it about?
 
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  • #2
What is your definition of the cross product?
 
  • #3
By the matrix definition of the cross product we have
[itex] \vec{a}\times \vec{b} \cdot \vec{c}
= \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_i & a_j & a_k \\ b_i & b_j & b_k \end{vmatrix} \cdot \vec{c}
= (\vec{i} \begin{vmatrix} a_j & a_k \\ b_j & b_k \end{vmatrix} -\vec{j} \begin{vmatrix} a_i & a_k \\ b_i & b_k \end{vmatrix} + \vec{k} \begin{vmatrix} a_i & a_j \\ b_i & b_j \end{vmatrix} ) \cdot \vec{c} \\
= (c_i \begin{vmatrix} a_j & a_k \\ b_j & b_k \end{vmatrix} -c_j \begin{vmatrix} a_i & a_k \\ b_i & b_k \end{vmatrix} + c_k \begin{vmatrix} a_i & a_j \\ b_i & b_j \end{vmatrix} )
= \begin{vmatrix} c_i & c_j & c_k \\ a_i & a_j & a_k \\ b_i & b_j & b_k \end{vmatrix} [/itex].

When [itex] \vec{c} = \vec{a} [/itex] or [itex] \vec{c} = \vec{b} [/itex] the determinant has two equal rows and becomes zero. This means the dot product is zero and the vectors are perpendicular.
 
  • #4
The cross product is the (up to multiplication by a constant) only product possible that takes two vectors to a third. It is also extremely useful to produce a vector perpendicular to two given vectors. All the time you have two vectors and need one perpendicular to them. Bam! Cross product done.
 
  • #5


The cross product is a mathematical operation that takes two vectors as input and produces a third vector that is perpendicular to both of the original vectors. This is why it is often referred to as the "vector product." The reason for this is rooted in the geometric properties of vectors and can be explained through the use of the right-hand rule.

The right-hand rule states that if you point your right thumb in the direction of the first vector and your fingers in the direction of the second vector, then the direction of the cross product will be perpendicular to both of these vectors. This is because the direction of the cross product is determined by the direction of rotation from the first vector to the second vector.

Now, let's consider the dot product of the cross product with one of the original vectors, say \vec{a}. The dot product is a scalar value that represents the projection of one vector onto another. In this case, the dot product represents the component of the cross product in the direction of \vec{a}. Since the cross product is perpendicular to both \vec{a} and \vec{b}, the dot product will always be zero. This means that the cross product is always perpendicular to the plane formed by the two original vectors.

So, to answer your question, the cross product is perpendicular because it is a vector that is determined by the direction of rotation between two vectors, and the dot product with either of these vectors will always be zero due to the perpendicularity of the cross product with the original vectors. This is not a random or arbitrary definition, but rather a geometric property of vectors that is fundamental to the concept of the cross product.
 

1. Why is the cross product perpendicular?

The cross product is perpendicular because it is defined as the vector that is perpendicular to both of the vectors being multiplied. This perpendicular vector is also known as the "normal" or "orthogonal" vector.

2. How is the cross product calculated?

The cross product is calculated using the determinant formula:

Cross Product Formula

where a and b are the two vectors being multiplied and i, j, and k are the unit vectors in the x, y, and z direction respectively.

3. What is the geometric interpretation of the cross product?

The cross product has a geometric interpretation as the area of the parallelogram formed by the two vectors being multiplied. The direction of the cross product is perpendicular to this area and can be determined using the right-hand rule.

4. Can the cross product be used in 2D?

No, the cross product is only defined in 3D. In 2D, the cross product would be equal to zero since there is no third dimension for the perpendicular vector to exist in.

5. How is the cross product used in physics and engineering?

The cross product is used in physics and engineering to calculate torque, angular momentum, and magnetic fields. It is also used in applications such as computer graphics and robotics to determine orientation and rotation.

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