Question

Find \(u \times v\) and show that it is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\). $$ \begin{array}{l} \mathbf{u}=\mathbf{i}+6 \mathbf{j} \\ \mathbf{v}=-2 \mathbf{i}+\mathbf{j}+\mathbf{k} \end{array} $$

Short answer

The cross product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is \(\mathbf{u} \times \mathbf{v} = 6\mathbf{i} - \mathbf{j} + 13\mathbf{k}\). This vector is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\) as the dot product of this vector with each of \(\mathbf{u}\) and \(\mathbf{v}\) is zero.

Step-by-step solution

Compute the cross product of \(\mathbf{u}\) and \(\mathbf{v}\)

The cross product of two vectors \(\mathbf{u} = (u_1,u_2,u_3)\) and \(\mathbf{v} = (v_1,v_2,v_3)\) in three-dimensional space is computed as follows: \( \mathbf{u} \times \mathbf{v} = (u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1) \). Plugging in the components of \(\mathbf{u}\) and \(\mathbf{v}\) into this formula, we get the vector \((6 \cdot 1 - 0 \cdot 1, 0 \cdot (-2) - 1 \cdot 1, 1 \cdot 1 - 6 \cdot (-2)) = (6, -1, 13)\). So, \(\mathbf{u} \times \mathbf{v} = 6\mathbf{i} - \mathbf{j} + 13\mathbf{k} \).

Check the orthogonality

A vector \(\mathbf{w}\) is orthogonal to \(\mathbf{u}\) and \(\mathbf{v}\) if the dot product of \(\mathbf{w}\) with \(\mathbf{u}\) and \(\mathbf{v}\) is zero, respectively. The dot product of two vectors, say, \(\mathbf{u} = (u_1,u_2,u_3)\) and \(\mathbf{v} = (v_1,v_2,v_3)\), is given by \( u_1v_1 + u_2v_2 + u_3v_3 \). Thus, if \(\mathbf{w} = \mathbf{u} \times \mathbf{v} = (6, -1, 13)\), then \(\mathbf{w} \cdot \mathbf{u} = 6 \cdot 1 + (-1) \cdot 6 + 13 \cdot 0 = 0 \), and \(\mathbf{w} \cdot \mathbf{v} = 6 \cdot (-2) + (-1) \cdot 1 + 13 \cdot 1 = 0 \). Both of these dot products are zero, hence \(\mathbf{w}\) is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\).

Key concepts

Vector Orthogonality

Understanding vector orthogonality is crucial when analyzing directions and planes in 3D space. Two vectors are said to be orthogonal if they meet at a right angle—that is, at 90 degrees. This seemingly basic concept has profound consequences. It helps in determining whether vectors are perpendicular, and this, in turn, plays a critical role in applications such as robotics, computer graphics, and physics.

But how can you tell if two vectors are orthogonal? The answer lies in the dot product. When the dot product of two vectors is zero, it confirms that they are orthogonal. Thus, checking for vector orthogonality is an essential step after finding the cross product of two vectors, as the resultant vector should be orthogonal to both original vectors. For instance, in the given exercise solution, verifying that the cross product, denoted by \(\mathbf{w}\), has a dot product of zero with the original vectors \(\mathbf{u}\) and \(\mathbf{v}\) affirms its orthogonality to both.

Dot Product

The dot product, also known as the scalar product, serves as a gateway to understanding vector interactions in mathematics. It is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. This number reveals valuable information about the vectors, such as their magnitude in a certain direction or, as previously mentioned, their orthogonality.

The dot product formula is simple yet powerful: for vectors \(\mathbf{u}\) and \(\mathbf{v}\) defined as \(\mathbf{u} = (u_1, u_2, u_3)\) and \(\mathbf{v} = (v_1, v_2, v_3)\), the dot product is given by \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3\). If this sum equals zero, we have confirmation that \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. The application of this operation in the problem not only shows the relation between vectors but also allows one to easily check the correctness of a cross product calculation.

Three-Dimensional Vectors

Three-dimensional vectors are the backbone of spatial analysis and are pivotal for anyone venturing into fields like engineering, physics, and computer graphics. They have both magnitude and direction and are not limited to a flat plane. A 3D vector is expressed as \(\mathbf{v} = (v_1, v_2, v_3)\) indicating its components along the x, y, and z axes respectively.

In our 3D world, the use of such vectors allows for detailed representation of points or directions in space. For example, the vectors \(\mathbf{u}\) and \(\mathbf{v}\) provided in the exercise represent directions in three-dimensional space. They can specify points for construction, model forces in physics, or indicate movement in space. When working with 3D vectors, operations like cross and dot products become especially meaningful as they help understand spatial relationships, such as the orthogonality demonstrated in the exercise.