Question

Find two orthogonal vectors that are both orthogonal to \(\mathbf{v}=\left[\begin{array}{l}1 \\ 2 \\ 0\end{array}\right]\).

Short answer

Two orthogonal vectors to \(\mathbf{v}\) are \([-2, 1, 0]^T\) and \([0, 0, 1]^T\).

Step-by-step solution

Understanding Orthogonality

The problem requires finding two vectors that are orthogonal to the given vector \(\mathbf{v}=[1, 2, 0]^T\). Orthogonality means the dot product of the vectors is zero.

Setting the Initial Vector

Consider a general vector \(\mathbf{u}=[x, y, z]^T\) that is orthogonal to \(\mathbf{v}\). The dot product of these vectors should be zero: \[ 1 \cdot x + 2 \cdot y + 0 \cdot z = 0 \]which simplifies to:\[ x + 2y = 0 \].

Solving for One Vector

To find the first orthogonal vector, set \(y = 1\) and solve for \(x\):\[ x + 2(1) = 0 \Rightarrow x = -2 \].Thus, one orthogonal vector is \(\mathbf{u}_1 = [-2, 1, 0]^T\).

Solving for Another Vector

To find a second orthogonal vector, choose \(z = 1\) and solve for \(x\) and \(y\). Since \(x + 2y = 0\), let's set \(y = 0\), so \(x=0\). This vector will be \(\mathbf{u}_2 = [0, 0, 1]^T\).

Verifying Orthogonality

Check if both vectors \(\mathbf{u}_1 = [-2, 1, 0]^T\) and \(\mathbf{u}_2 = [0, 0, 1]^T\) are orthogonal to \(\mathbf{v}\):- For \(\mathbf{u}_1\), check: \[ 1(-2) + 2(1) + 0(0) = 0 \]- For \(\mathbf{u}_2\), check:\[ 1(0) + 2(0) + 0(1) = 0 \]Both results are zero, confirming the orthogonality.

Key concepts

Dot Product

The dot product is a key concept when working with vectors. It's an operation that takes two vectors and returns a scalar (a single number). This scalar helps determine important properties like length and angle between vectors. To compute the dot product for two vectors \( \mathbf{a} = [a_1, a_2, a_3] \) and \( \mathbf{b} = [b_1, b_2, b_3] \), use the formula:
\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\]
Here's why the dot product is helpful:

• If the dot product of two vectors is zero, the vectors are orthogonal (at right angles to each other).
• If the dot product is positive, they point in a similar direction.
• If the dot product is negative, they point in opposite directions.

Understanding how to calculate and interpret the dot product is crucial in vector analysis.

Orthogonality

Orthogonality is a concept that means two vectors are perpendicular to one another. This occurs when their dot product equals zero. Imagine two lines crossing at a perfect right angle—this is essentially what orthogonal vectors do in space.
When trying to determine if vectors \( \mathbf{u} = [u_1, u_2, u_3] \) and \( \mathbf{v} = [v_1, v_2, v_3] \) are orthogonal, simply check:
\[ u_1v_1 + u_2v_2 + u_3v_3 = 0 \]
In applications:

• Orthogonality is essential in fields like computer graphics and machine learning for simplifying complex problems.
• Knowing vectors are orthogonal helps in tasks like constructing coordinate systems.

Recognizing and using orthogonality enables us to simplify and solve many mathematical problems.

Vector Operations

Vector operations involve several mathematical processes that enable us to manipulate and interpret vectors effectively. Common operations include addition, subtraction, and scaling (multiplying by a constant).
For example, adding two vectors \( \mathbf{a} = [a_1, a_2, a_3] \) and \( \mathbf{b} = [b_1, b_2, b_3] \) results in:
\[ \mathbf{a} + \mathbf{b} = [a_1 + b_1, a_2 + b_2, a_3 + b_3] \]
Here's why understanding vector operations is valuable:

• They allow us to perform movements and rotations in physics and engineering.
• In computer graphics, they help create and animate objects in a digital environment.
• Operations like scaling change the size of a vector while maintaining its direction.

Mastering these operations enables us to work with vectors in practical and theoretical contexts with ease.