What Are the Unit Vectors Perpendicular to Both a Vector ab and a-b?

In this article, we will explore the process of finding unit vectors that are perpendicular to both ab and a - b using the cross product method. This exploration will involve step-by-step calculations and the concept of orthogonal vectors in R3.

Introduction to the Vectors and the Cross Product

Given the vectors a i j k and b i 2j 3k, we first establish ab 2i 3j 4k and a - b -j - 2k.

Calculating the Cross Product

The first step in finding the vector that is perpendicular to both ab and a - b involves calculating their cross product. The cross product of ab and a - b can be done as follows:

Step 1: Set Up the Determinant

The cross product of two vectors can be found by setting up the determinant of a 3x3 matrix:

[ begin{vmatrix} i j k 2 3 4 0 -1 -2 end{vmatrix} ]

[ i(3(-2) - 4(-1)) - j(2(-2) - 4(0)) k(2(-1) - 3(0)) ]

[ -2i 4j - 2k ]

Step 2: Calculate the Magnitude

The magnitude of the resulting vector is calculated as follows:

[ left| -2i 4j - 2k right| sqrt{(-2)^2 4^2 (-2)^2} sqrt{4 16 4} sqrt{24} ]

[ 2sqrt{6} ]

Step 3: Normalize the Vector

The unit vector is then found by dividing the cross product by its magnitude:

[ hat{u} frac{-2i 4j - 2k}{2sqrt{6}} -frac{1}{sqrt{6}}i frac{2}{sqrt{6}}j - frac{1}{sqrt{6}}k ]

[ -frac{1}{sqrt{6}}i frac{2}{sqrt{6}}j - frac{1}{sqrt{6}}k ]

Understanding the Perpendicularity Condition

Given that the vectors a and b lie in the same plane, the cross product a x b itself is a vector perpendicular to both ab and a - b. This can be verified by taking the dot product of the resulting vector with both ab and a - b.

Units Vectors Along Any Non-Zero Vector

Any unit vector along a non-zero vector u can be obtained by dividing u by its magnitude, u. We will calculate the vectors orthogonal to a and b, and find a common vector orthogonal to both:

Orthogonal Vectors to a and b

For a vector to be orthogonal to a i j k, it must satisfy:

[ 2x 3y 5z 0 ]

For a vector to be orthogonal to b i 2j 3k, it must satisfy:

[ -y - 2z 0 ]

By solving these equations, we find that any common vector orthogonal to both a and b must be of the form:

[ (x, -4x, 2x) ]

Thus, any multiple of 1 - 4j 2k is perpendicular to both a and b.

Unit Vector Calculation

The magnitude of 1 - 4j 2k is:

[ sqrt{1^2 (-4)^2 2^2} sqrt{1 16 4} sqrt{21} ]

Thus, the unit vector is:

[ hat{u} frac{1}{sqrt{21}} - 4j 2k ]

Conclusion and Applications

The concept of perpendicular vectors and their unit forms is fundamental in various fields, including physics, engineering, and computer graphics. Understanding these concepts through the cross product and dot product can provide deeper insights into vector operations and their applications in real-world scenarios.

Key Takeaways

The cross product helps in identifying vectors that are perpendicular to a pair of given vectors. The unit vector of a vector is found by dividing the vector by its magnitude. Perpendicular vectors find applications in various scientific and engineering fields.

Keywords: Cross Product, Perpendicular Vectors, Unit Vectors