Question

Give an example of three vectors in ${\mathrm{ℝ}}^3$that form a right-handed triple. Explain how you can use the same three vectors to form a left-handed triple.

Short answer

The example of a right-handed triple is $i,j,k$.

The same three vectors form a left-handed triple if W lies on the side of the plane through O, U, V does not indicated by your second [middle] finger.

Step-by-step solution

Step 1. Given Information 

Give an example of three vectors in ${\mathrm{ℝ}}^3$ that form a right-handed triple. Explain how you can use the same three vectors to form a left-handed triple.

Step 2. Right-handed and left-handed triples

Suppose now that u, v, w are not coplanar. Pick an origin $O$ in 3-space, and define U, V , W by the condition that $\overrightarrow{OU},\overrightarrow{OV},\overrightarrow{OW}$ represent u, v, w respectively. Using your right hand put your thumb in the direction of u, and your first [index] finger in the direction of v.

Step 3. If W lies on the side of the plane through O, U, V indicated by your second [middle] finger, we call u, v, w a right-handed triple; otherwise it is a left-handed triple.

The example of a right-handed triple is$i,j,k.$

The same three vectors form a left-handed triple if W lies on the side of the plane through O, U, V does not indicated by your second [middle] finger.