# Question on the right hand rule

Say I'm taking the cross product of vectors $$a$$ and $$b$$. Say that $$b$$ is totally in the $$z$$ direction and has length $$7$$, so $$b = 7k$$. Say that $$a$$ is in the $$xy$$-plane with positive coefficients, $$a = 3x + 4y$$.

I want to understand the sign of the components of $$a \times b$$ using the right hand rule. Now, surele since $$a \times b$$ is orthogonal to both $$a$$ and $$b$$, it's $$z$$ component will be zero. But will the $$x$$ and $$y$$ components be positive or negative, and how can i see this with the right hand rule? Thank you for your time

• Open the fingers on your right hand, point towards $a$ by your thumb and towards $b$ by your index finger and then the middle finger will show you the direction of $a\times b$. – Berci Dec 2 '20 at 18:36

Take a piece of paper, and place a pencil on top of it. That pencil would represent the $$z$$-axis.

Your thumb should follow the direction of vector $$a$$, your index finger is pointing towards the $$b$$ direction and you will notice that your midddle finger should be pointing to the fourth quadrant, hence the $$x$$ coordinate is positive and the $$y$$ coordinate is negative.

• thanks, i misread $a$ and $b$. – Siong Thye Goh Dec 2 '20 at 19:49

Remember that $$a$$ points in direction $$(3,4)$$ which is little above $$y=x$$ ($$45^{\circ}$$) line.

Open your palm so your fingers point in direction of a. Now curl your four fingers towards b. Your stretched out thumb will point in direction of $$a \times b$$.

Easy to see $$x$$-component is positive and $$y$$-component negative. More precisely, the components will be proportional to $$(4,-3)$$ so that $$(3,4)\cdot(4,-3)=0$$

Yep, the cross product will have no z component. The cross product of vector in the xy plane with a vector perpendicular to the plane has the effect of rotating the vector plus or minus 90 degrees then scaling it. If the perpendicular vector has a positive component, the plane's vector rotates clockwise, otherwise couter-clockwise, by the right hand rule.

This applies just as well to the unit vectors. So the positive $$\hat{y}$$ unit vector becomes the positive $$\hat{x}$$ unit vector.

Rotating a vector doesn't necessarily change the sine of its component. $$<\frac{-\sqrt{2}}{2},\frac{\sqrt{2}}{2}>$$ doesn't change the sign of its y component upon taking the cross product.