Understanding Vector Addition to Achieve a Unit Vector Along the Z-Axis

In the realm of vector mathematics, vector addition plays a crucial role in achieving specific outcomes, such as obtaining a unit vector along a particular axis. This article will guide you through the process of finding a vector that needs to be added to the given vectors -3i 2j and i -j 3k, so that the resultant vector is a unit vector along the z-axis. Let's delve into the detailed steps and mathematical reasoning behind this process.

Understanding the Problem

We are given two vectors:

A -3i 2j k B i - j 3k

The question is: What vector V must be added to these two vectors so that the resultant vector is a unit vector along the z-axis?

Step-by-Step Solution

1. **Calculate the resultant vector R of A and B:** [ R A - B (-3i 2j k) - (i - j 3k) ] Combining like terms:

[ R -3i - i 2j j k - 3k ] [ R -2i 1j (-2)k ]

2. **Identify the required resultant vector U:** A unit vector along the z-axis is represented as: [ U 0i 0j 1k ] Therefore, the components of the required vector V must be determined such that the resultant vector R becomes U when V is added to R.

Mathematical Derivation

Let the vector to be added be ( V x i y j z k ). The new resultant vector after adding V will be:

R R V ( - 2 i 1 j - 2 k ) x i y j z k - 2 i 1 j - 2 k x i y j z k

We want the resultant vector to be a unit vector along the z-axis (0i 0j 1k). Therefore, we set the components equal:

( -2 x 0 ) ( 1 y 0 ) ( -2 z 1 )

Solving these equations, we get:

From ( -2 x 0 ): ( x 2 ) From ( 1 y 0 ): ( y -1 ) From ( -2 z 1 ): ( z -3 )

Hence, the vector V that must be added is:

V 2 i - 1 j - 3 k

Conclusion

In summary, the vector that must be added to -3i 2j and i -j 3k so that the resultant vector is a unit vector along the z-axis is:

boxed{2i - j - 3k}

This step-by-step approach ensures that you can solve similar problems by understanding vector addition and the properties of unit vectors. For more detailed explanations and further exercises, consider exploring advanced vector algebra resources.