Register to reply 
Why is the cross product perpendicular? 
Share this thread: 
#1
Feb1913, 02:43 PM

P: 47

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? 


#3
Feb2013, 01:25 PM

P: 22

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
Feb2113, 01:51 AM

HW Helper
P: 2,264

Why is the cross product perpendicular?
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.



Register to reply 
Related Discussions  
Given one cross product, find another cross product  Calculus & Beyond Homework  1  
Angle between 2 vectors using 1) Dot product and 2) cross product gives diff. answer?  Calculus & Beyond Homework  8  
Perpendicular vector using dot not cross product.  Introductory Physics Homework  4  
Cross product and dot product of forces expressed as complex numbers  Introductory Physics Homework  4  
What does crosssections perpendicular mean?  Calculus  2 