Question

Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal parallel, or neither. $$ \begin{array}{l} \mathbf{u}=-\frac{1}{3}(\mathbf{i}-2 \mathbf{j}) \\ \mathbf{v}=2 \mathbf{i}-4 \mathbf{j} \end{array} $$

Short answer

The vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal and not parallel.

Step-by-step solution

Calculation of Dot Product

The dot product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is calculated as follow: \(\mathbf{u} \cdot \mathbf{v} = (-1/3)i \cdot 2i + (-1/3)(-2)j \cdot (-4)j = -2/3 + 2/3 = 0\)

Checking Orthogonality

Since the dot product of the vectors \(\mathbf{u}\) and \(\mathbf{v}\) is zero, we can conclude that \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal.

Checking Parallelism

Two vectors are parallel if one is a scalar multiple of the other. While it's true that the \(\mathbf{v}\) could be written as -3 times \(\mathbf{u}\), we dont need to check for parallelism after finding the vectors to be orthogonal. Since being orthogonal and parallel are mutually exclusive properties in vector space. Hence, vectors \(\mathbf{u}\) and \(\mathbf{v}\) are not parallel.

Key concepts

Dot Product

Understanding the dot product is essential in vector mathematics and provides valuable insight into the relationship between two vectors. The dot product, also known as the scalar product, involves multiplying corresponding components of two vectors and then summing those products. Expressed mathematically, if we have two vectors, \( \mathbf{a} = a_i \mathbf{i} + a_j \mathbf{j} \) and \( \mathbf{b} = b_i \mathbf{i} + b_j \mathbf{j} \) in two-dimensional space, their dot product \( \mathbf{a} \cdot \mathbf{b} \) is given by: \[ \mathbf{a} \cdot \mathbf{b} = a_i b_i + a_j b_j \
\]In the context of the given exercise, we calculate the dot product of vectors \(\mathbf{u} \) and \(\mathbf{v} \) to determine a specific relationship between them.

Interpreting the Dot Product

The result of the dot product can tell us more than just the magnitude. If the dot product is zero, it indicates a special relationship known as orthogonality. This relationship is what gives us the cue to consider whether vectors are orthogonal. Non-zero positive or negative results can indicate other vector relationships—such as if vectors are parallel or if they have an acute or obtuse angle between them.

Vector Orthogonality

When two vectors are orthogonal, they intersect at a right angle (90 degrees) to each other. This is a fundamental concept in geometry and vector algebra and has significant implications in fields such as computer graphics, physics, and more.
To identify if two non-zero vectors are orthogonal, we examine their dot product. If the dot product equals zero, this confirms orthogonality, indicating that they are perpendicular.

Application in Exercises

In the aforementioned exercise, after we calculate the dot product and find it to be zero, we can immediately conclude that vectors \(\mathbf{u} \) and \(\mathbf{v} \) are orthogonal. This is a crucial step in the solution process, as it guides us away from further unnecessary calculations, such as checking for parallelism, when we already know the vectors are perpendicular.

Parallel Vectors

Parallel vectors are vectors that run in the same direction or the exact opposite direction but do not intersect. To determine if two vectors are parallel, one must be a scalar multiple of the other, indicating that they are aligned along the same line or plane.

Identifying Parallel Vectors

To identify parallel vectors, we look for a constant that, when multiplied by one vector, equals the other vector. For vectors in two-dimensional space, \(\mathbf{a} \) is parallel to \(\mathbf{b} \) if there exists a scalar \( k \) such that \[ k\mathbf{a} = \mathbf{b} \
\]However, in the exercise, once we understand that the dot product is zero and the vectors are orthogonal, we do not need to check parallelism because a vector cannot be both orthogonal and parallel to another vector simultaneously. The properties of orthogonality and parallelism are mutually exclusive in the context of non-zero vectors.