Final answer:
A vector in three-dimensional space can be expressed as a combination of its components along the unit vectors i, j, and k. The cross product of any two unit vectors yields either the third unit vector or its negative, depending on their order in the product.
Step-by-step explanation:
To express a vector in terms of the unit vectors i, j, and k, you need to apply the rules for vector cross products and dot products. The cross product of two unit vectors in three-dimensional space can be determined by the cyclic order of i, j, and k. When the unit vectors appear in this order, their cross product will yield the third unit vector. If the order is reversed, the result will be the negative of the third unit vector.
For example, using the rules described:
i x i = 0, j x j = 0, k x k = 0
i x j = k, j x k = i, k x i = j
j x i = -k, k x j = -i, i x k = -j
The magnitudes of all unit vectors are one (|i| = |j| = |k| = 1), and they are mutually orthogonal, forming a right-handed coordinate system. The dot product of a vector with a unit vector yields the component of the vector in the direction of the unit vector. For instance, a vector A with components Ax, Ay, and Az in three dimensions can be expressed as A = Axi + Ayj + Azk.