Final answer:
The cross product k × (i - 3j) is calculated by using the properties of cross products between unit vectors, resulting in the vector 3i - j.
Step-by-step explanation:
The cross product k × (i - 3j) using cross product properties leads to k × i = -j and k × -3j = -3(-i), given that j × k = i and due to anti-commutativity (k × i = -i × k). Thus, adding the two cross products, we obtain the final vector as -j + 3i.
Direct answer in 2 lines: The cross product k × (i - 3j) is 3i - j.
When finding the cross product k with the vector i - 3j, we must consider the cross products of the individual unit vectors involved. According to the properties of the cross product, the cross product of any two perpendicular unit vectors results in the third unit vector, respecting the right-hand rule and cyclic order. Considering that k × i = -j and k × j = i, and noting the anti-commutative property (k × -3j = -3k × j), we must reverse the direction for the latter product. Combining these results using the distributive property, we get k × (i - 3j) = k × i - 3(k × j) = -j + 3i, arriving at the cross product being 3i - j, a vector laying in the xy-plane.