Final answer:
To find (i x j) x k using properties of cross products, we utilize the cyclic order rule and note that the cross product of a unit vector with itself is zero. Hence, (i x j) x k equals 0.
Step-by-step explanation:
To find the vector product (i x j) x k without using determinants and by using properties of cross products, we need to apply the rules for cross-multiplication of unit vectors. First, note that the cross product of two different unit vectors following a cyclic order is the third unit vector in positive orientation. For example, i x j = k because they follow the cyclic order. However, when they do not appear in the cyclic order, the result is the unit vector in a negative orientation. In this case, (i x j) x k does not follow the cyclic order, hence the result will have a negative sign.
Applying the cyclic order, since i x j = k, we then have k x k. Now, according to the properties of cross products, any unit vector crossed with itself is zero. Thus,
k x k = 0. Therefore, (i x j) x k also equals to 0.
Remembering that the corkscrew rule helps in determining the direction of the cross product, and recognizing that two of the same unit vectors crossed result in a null vector, are important while evaluating such expressions.