Question

Find \(u \times v\) and show that it is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\). $$ \begin{array}{l} \mathbf{u}=\langle-1,1,2\rangle \\ \mathbf{v}=\langle 0,1,0\rangle \end{array} $$

Short answer

The cross product \(\mathbf{u} \times \mathbf{v} = \langle-2,0,1\rangle\), and it is indeed orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\) as their dot products are zero.

Step-by-step solution

Calculate \(\mathbf{u} \times \mathbf{v}\)

For \(\mathbf{u}=\langle-1,1,2\rangle\) and \(\mathbf{v}=\langle 0,1,0\rangle\), the cross product is calculated as follows: \(\mathbf{u} \times \mathbf{v} = \left|\begin{matrix}i & j & k \-1 & 1 & 2 \0 & 1 & 0 \\end{matrix}\right| = i \left|\begin{matrix}1 & 2 \1 & 0 \\end{matrix}\right| - j \left|\begin{matrix}-1 & 2 \0 & 0 \\end{matrix}\right| + k \left|\begin{matrix}-1 & 1 \0 & 1 \\end{matrix}\right| = i(0-2) - j(0-0) + k(1-0) = -2i + k = \langle-2,0,1\rangle.\)

Check orthogonal to \(\mathbf{u}\)

A vector is orthogonal to another if their dot product is zero. For \(\mathbf{u}=\langle-1,1,2\rangle\) and \(\mathbf{u} \times \mathbf{v}=\langle-2,0,1\rangle\), \(\mathbf{u}.\mathbf{u} \times \mathbf{v} = -1*-2 + 1*0 + 2*1 = 0.\)

Check orthogonal to \(\mathbf{v}\)

Continuing in the same manner for \(\mathbf{v}=\langle 0,1,0\rangle\) and \(\mathbf{u} \times \mathbf{v}=\langle-2,0,1\rangle\), \(\mathbf{v}.\mathbf{u} \times \mathbf{v} = 0*-2 + 1*0 + 0*1 = 0.\)

Key concepts

Orthogonal Vectors

When two vectors are orthogonal, they meet at a right angle (90 degrees) to each other. This is a vital concept in both mathematics and physics, as it often signifies perpendicularity and has important implications in various calculations, such as projections and forces.

To understand and identify orthogonality between vectors, we use the dot product (also known as the scalar product). If the dot product of two non-zero vectors is zero, they are considered orthogonal. For example, in the solved problem, after calculating the cross product, we can use the dot product to prove that the resulting vector is orthogonal to the original vectors. The zero results of the dot products in Step 2 and Step 3 confirmed the orthogonality.

Vector Multiplication

Vector multiplication can be performed in several ways, among which the cross product and dot product are the most common methods.

The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space. The result of the cross product is a vector that is orthogonal to the plane formed by the two input vectors. To calculate the cross product, we use the determinant of a matrix that includes the unit vectors along with the components of the vectors involved. The result is a new vector with unique components, as demonstrated in Step 1 of the problem.

Cross Product Properties

• The resultant vector is perpendicular to the plane containing the initial vectors.
• It adheres to the right-hand rule, which helps determine the direction of the resultant vector.
• The magnitude of the cross product relates to the area of the parallelogram with sides represented by the original vectors.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. This operation is fundamental when dealing with vector calculus and geometry.

Calculating a dot product is straightforward: you multiply corresponding components of the vectors and then sum these products. The formula for the dot product \(\mathbf{a} \cdot \mathbf{b}\) of two vectors \(\mathbf{a} = \langle a_1, a_2, a_3 \rangle\) and \(\mathbf{b} = \langle b_1, b_2, b_3 \rangle\) is given by \(a_1b_1 + a_2b_2 + a_3b_3\).

Applications of Dot Product

• Projection: The dot product can help find the projection of one vector onto another.
• Orthogonality check: As explained earlier, a dot product of zero indicates that two vectors are orthogonal.
• Angle measurement: It can be used to find the cosine of the angle between two vectors, which gives us information about their directional relationship.

For instance, in the given exercise, the dot product is used in steps 2 and 3 to confirm that the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal to their cross product \(\mathbf{u} \times \mathbf{v}\).