Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. The vectors \(\mathbf{u}=\langle 0,0\rangle\) and \(\mathbf{v}=\langle-12,6\rangle\) are orthogonal.

Step-by-Step Solution

Answer

Yes, the vectors are orthogonal because their dot product equals zero and one of the vectors is a zero vector.

Step 1: Find the Dot Product of the Vectors

To assess orthogonality, we begin by computing the dot product of the given vectors. The dot product of two vectors \( \mathbf{u}=\langle u_1,u_2 \rangle \) and \( \mathbf{v}=\langle v_1,v_2 \rangle \) is computed as \( \mathbf{u} \cdot \mathbf{v} = u_1*v_1 + u_2*v_2 \). Therefore, the dot product of \( \mathbf{u} \) and \( \mathbf{v} \) is \( 0*(-12) + 0*6 = 0 \).

Step 2: Check the Dot Product

Vectors are considered orthogonal if their dot product is zero. Here, the dot product of \( \mathbf{u} \) and \( \mathbf{v} \) is 0, which indicates that they could be orthogonal.

Step 3: Additional Verification

While the dot product being zero is generally a strong indicator of orthogonality, it's important to remember that the zero vector (in this case, \( \mathbf{u} \)) is orthogonal to any vector. Therefore, this doesn't necessarily confirm that a non-zero vector is orthogonal. However, in this case, given that one of the vectors is the zero vector, our earlier conclusion holds. The vectors \( \mathbf{u}=\langle 0,0\rangle \) and \( \mathbf{v}=\langle-12,6\rangle \) are indeed orthogonal.

Key Concepts

Dot Product

Understanding the dot product is crucial for many aspects of vector mathematics, including the determination of if two vectors are orthogonal. The dot product is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation is deeply tied with concepts in geometry, physics, and engineering.

In mathematical terms, the dot product of two vectors, let’s represent them as \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) in two-dimensional space, is calculated as \( \mathbf{a} \cdot \mathbf{b} = a_1 \times b_1 + a_2 \times b_2 \) The result is a scalar, which means it is a single number, not a vector.

In the context of our exercise, the vectors given are \( \mathbf{u}=\langle 0,0\rangle \) and \( \mathbf{v}=\langle-12,6\rangle \) and their dot product equals 0, which leads us to explore the conditions for vector orthogonality.

Vector Orthogonality

Vector orthogonality is a fundamental concept in vector algebra with practical implications across various fields such as computer graphics, physics, and statistics. Two vectors are defined as orthogonal if the angle between them is 90 degrees; this translates to them being perpendicular to each other. A convenient method to test for orthogonality is through the dot product, as mentioned previously.

If the dot product of two non-zero vectors is zero, this implies that the vectors are orthogonal, for example, \( \mathbf{a} \cdot \mathbf{b} = 0 \) suggests that \( \mathbf{a} \) and \( \mathbf{b} \) are perpendicular. In our textbook exercise, we discover that the dot product is indeed zero, but there's an extra layer. One of our vectors is a zero vector which, by definition, is orthogonal to any other vector. It's crucial to note that orthogonality can translate into concepts of independence, where orthogonal vectors can represent different, independent directions in space.

Zero Vector

The zero vector, often denoted as \( \mathbf{0} \) or \( \mathbf{u}=\langle 0,0\rangle \) in two dimensions, plays a unique role in vector mathematics. This vector, which contains all zero components, acts as the additive identity in vector space. Any vector added to the zero vector will result in the original vector. Additionally, the zero vector provides a reference for vector orthogonality.

While any vector is orthogonal to the zero vector, the zero vector does not provide information about the spatial orientation of other vectors. This fact is relevant in the exercise we're analyzing. To restate, while it is true that the dot product of \( \mathbf{u} \) and \( \mathbf{v} \) being zero indicates orthogonality, it's the zero vector's intrinsic property to be orthogonal to all vectors that confirms the statement without further spatial consideration. As a result, when working with zero vectors, we don't interpret orthogonality as an indicator of independent directions.