Problem 38 - APEX Calculus

Question

Find a unit vector orthogonal to both \(\vec{u}\) and \(\vec{v} .\) \(\vec{u}=\langle 1,-2,1\rangle, \quad \vec{v}=\langle-2,4,-2\rangle\)

Step-by-Step Solution

Answer

No such unit vector exists; \(\vec{u}\) and \(\vec{v}\) are parallel.

Step 1: Find the Cross Product

To find a unit vector orthogonal to both \(\vec{u}\) and \(\vec{v}\), we first need to find the cross product \(\vec{u} \times \vec{v}\). The cross product of two vectors in three dimensions is given by:\[\vec{u} \times \vec{v} = \langle u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1 \rangle\]Applying this formula to \(\vec{u} = \langle 1,-2,1 \rangle\) and \(\vec{v} = \langle -2,4,-2 \rangle\), we calculate each component:- First component: \((-2)(-2) - (1)(4) = 4 - 4 = 0\)- Second component: \((1)(-2) - (1)(-2) = -2 + 2 = 0\)- Third component: \((1)(4) - (-2)(-2) = 4 - 4 = 0\)The cross product is \(\langle 0, 0, 0 \rangle\).

Step 2: Check for Zero Cross Product

The cross product of \(\vec{u}\) and \(\vec{v}\) is \(\langle 0, 0, 0 \rangle\), which is a zero vector. This implies that \(\vec{u}\) and \(\vec{v}\) are parallel or one is a scalar multiple of the other. In such a case, we cannot find a unit vector orthogonal to both \(\vec{u}\) and \(\vec{v}\).

Step 3: State Conclusion

Since the cross product is the zero vector, there is no unit vector orthogonal to both \(\vec{u}\) and \(\vec{v}\), as they are parallel or linearly dependent.

Key Concepts

Cross Product

The cross product is a fundamental operation in vector calculus that allows us to find a vector that is orthogonal (or perpendicular) to two given vectors. It is only applicable in three-dimensional space and is denoted by the symbol \(\times\). The cross product of two vectors \(\vec{a}\) and \(\vec{b}\) is written as \(\vec{a} \times \vec{b}\). To compute this, you use the determinant of a matrix formed by the unit vectors \(\hat{i}, \hat{j}, \hat{k}\) and the components of the vectors \(\vec{a}\) and \(\vec{b}\).

• First Component: \(a_2b_3 - a_3b_2\)
• Second Component: \(a_3b_1 - a_1b_3\)
• Third Component: \(a_1b_2 - a_2b_1\)

Each component is calculated based on the specific elements of the vectors involved. An important aspect of this operation is that if the cross product results in the zero vector, it indicates that the two vectors are parallel or collinear, meaning one is a scalar multiple of the other. In such cases, they do not form a plane, and hence, no perpendicular vector exists between them.

Orthogonal Vectors

Orthogonal vectors are vectors that meet at a right angle, or 90 degrees. This means that they have a dot product of zero. The dot product is another vector operation that provides a scalar and is calculated as \(\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3\).
When the dot product equals zero, it signifies orthogonality. The significance of finding a vector orthogonal to two known vectors lies in applications such as finding a normal to a plane, which can be invaluable in fields of physics and engineering.

• Orthogonality implies perpendicular direction.
• The cross product of any two vectors is an attempt to find a vector orthogonal to both.
• If the cross product is zero, no unique orthogonal vector exists.

Understanding orthogonality is crucial in vector analysis, as it allows for simplifications and insights in complex problems.

Unit Vector

A unit vector is a vector with a magnitude of one. It serves to indicate direction without affecting magnitude. To convert any vector into a unit vector, you divide the vector by its magnitude. Given a vector \(\vec{a} = \langle a_1, a_2, a_3 \rangle\), the unit vector \(\hat{a}\) is calculated as follows:\[ \hat{a} = \frac{\vec{a}}{\| \vec{a} \|} \]Where \(\| \vec{a} \|\) represents the magnitude of the vector, computed by:\[ \| \vec{a} \| = \sqrt{a_1^2 + a_2^2 + a_3^2} \]Unit vectors are essential in mathematics and physics because they provide standard directions for referencing, while maintaining unit length.

• Used extensively in calculations and transformations.
• Coordinates direction without scaling the original vector.
• Facilitates understanding of vector fields and changes in vector functions.

The concept of unit vectors simplifies many calculations by standardizing direction to a single magnitude.