cross product question

**ave** · May 21st 2009, 12:59 AM

once I have determined the cross product of an angle, how do I tell if its facing up or down- ie a negative or positive angle- especially if im dealing with a plane that could be lying in any direction in 3d space

**HallsofIvy** · May 22nd 2009, 06:58 AM

Originally Posted by ave

once I have determined the cross product of an angle, how do I tell if its facing up or down- ie a negative or positive angle- especially if im dealing with a plane that could be lying in any direction in 3d space

The cross product of an angle? Do you mean the cross product of two vectors lying at a given angle? And what, exactly, do you mean by "a negative or positive angle"? The direction of a line or vector in 3 space requires 3 angles. Since you specify "up or down", I would say just look at the z-component. If it is positive, up, negative, down.

**ave** · May 22nd 2009, 07:51 AM

Originally Posted by HallsofIvy

The cross product of an angle? Do you mean the cross product of two vectors lying at a given angle? And what, exactly, do you mean by "a negative or positive angle"? The direction of a line or vector in 3 space requires 3 angles. Since you specify "up or down", I would say just look at the z-component. If it is positive, up, negative, down.

unfortunately because the vectors lying at a given angle as you correctly put it, could be anywhere in 3d space, the cross product doesnt necessarily lie on the z axis, that is exactly the problem....

what I mean by a negative or positive angle is- lets assume the first vector is x=1, y=0, and the other vector is lying at y=1, x=0, that would be +90, where as if the second vector is at y=-1, x=0 , that would be -90 degrees

now using the right hand rule I can tell which direction the cross product is going to face, unfortunately I am writing a computer program that plots points, so the right hand rule doesnt work for me... I want to know how to do this mathematically

**HallsofIvy** · May 22nd 2009, 10:15 AM

Originally Posted by ave

unfortunately because the vectors lying at a given angle as you correctly put it, could be anywhere in 3d space, the cross product doesnt necessarily lie on the z axis, that is exactly the problem....

I didn't assume the cross product was lying in the z-axis. I said "look at the z component of the cross product.

what I mean by a negative or positive angle is- lets assume the first vector is x=1, y=0, and the other vector is lying at y=1, x=0, that would be +90, where as if the second vector is at y=-1, x=0 , that would be -90 degrees

now using the right hand rule I can tell which direction the cross product is going to face, unfortunately I am writing a computer program that plots points, so the right hand rule doesnt work for me... I want to know how to do this mathematically

The cross product of $x_1\vec{i}+ y_1\vec{j}+ z_1\vec{k}$ and $x_2\vec{i}+ y_2\vec{j}+ z_2\vec{k}$ is $(y_2z_3- z_2y_3)\vec{i}+ (x_2z_1- z_2x_1)\vec{j}+ (x_1y_2- y_1x_2)\vec{k}$ . It is pointing "up" if $x_1y_2- y_1x_2$ is positive and "down" if negative.

**ave** · May 22nd 2009, 12:38 PM

Originally Posted by HallsofIvy

I didn't assume the cross product was lying in the z-axis. I said "look at the z component of the cross product.

The cross product of $x_1\vec{i}+ y_1\vec{j}+ z_1\vec{k}$ and $x_2\vec{i}+ y_2\vec{j}+ z_2\vec{k}$ is $(y_2z_3- z_2y_3)\vec{i}+ (x_2z_1- z_2x_1)\vec{j}+ (x_1y_2- y_1x_2)\vec{k}$ . It is pointing "up" if $x_1y_2- y_1x_2$ is positive and "down" if negative.

oh ok I get it now- thanks for the help!! I really appreciate it

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