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The cross product is a fundamental operation in vector algebra. It is used to find a vector that is perpendicular to two given vectors in three-dimensional space. Let’s look at how this is done step-by-step.

A cross product between two vectors \(\mathbf{a} = \langle a_{1}, a_{2}, a_{3} \rangle\) and \(\mathbf{b} = \langle b_{1}, b_{2}, b_{3} \rangle\) results in a new vector \(\mathbf{c} = \langle c_{1}, c_{2}, c_{3} \rangle\). The components of \(\mathbf{c}\) are found using:
\[c_{1} = a_{2}b_{3} - a_{3}b_{2},\]
\[c_{2} = a_{3}b_{1} - a_{1}b_{3},\]
\[c_{3} = a_{1}b_{2} - a_{2}b_{1}.\]

The cross product is only defined in three dimensions. This makes it especially useful in many areas of physics and engineering where 3D space is considered.

In the solution above, we found two orthogonal vectors by strategically choosing basis vectors and performing cross products. By using the cross product, we made sure to find vectors that are not only perpendicular to \(\langle 0,1,1\rangle\) but also to each other.

Basis vectors are the building blocks of vector spaces. In three-dimensional space, the standard basis vectors are \(\langle 1,0,0 \rangle\), \(\langle 0,1,0 \rangle\), and \(\langle 0,0,1 \rangle\).

These vectors are mutually orthogonal and they form the framework upon which any vector in 3D space can be constructed. You can think of them as axes in a coordinate system, providing directions along the x, y, and z axes.

When solving problems involving orthogonality, basis vectors can be invaluable. They offer a simple way to perform operations like cross products due to their straightforward components. This simplicity helps avoid errors when calculating more complex vector relations.

In the solution provided, we used the basis vector \(\langle 1,0,0 \rangle\) initially. This choice allowed us to find an orthogonal vector quickly without getting a zero vector, which would provide no use in finding orthogonality.

Vector algebra is a field of mathematics that deals with vectors and operations on them. This subject provides the tools needed to work with vectors in various dimensions.

Basic operations include addition, subtraction, and scalar multiplication. Here vectors can be added or subtracted component-wise, producing another vector.

A key operation within vector algebra is the dot product. It results in a scalar and can tell us if two vectors are orthogonal. If their dot product is zero, the vectors are orthogonal.

In this exercise, however, the emphasis was on the cross product because it directly returns an orthogonal vector. Vector algebra underpins many areas of science and engineering, providing a rich framework to handle physical quantities like forces and velocities, which are naturally vector quantities.

The step-by-step solution above showcases an application of vector algebra to find orthogonal vectors efficiently, highlighting how these mathematical operations can be used to solve real-world problems.