Question

Find \(u \times v\) and show that it is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\). $$ \begin{array}{l} \mathbf{u}=\langle-10,0,6\rangle \\ \mathbf{v}=\langle 7,0,0\rangle \end{array} $$

Short answer

The cross product of vectors \( \mathbf{u} \) and \( \mathbf{v} \) is \( \mathbf{w} = \langle 0, 70, 0 \rangle \), which is orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \).

Step-by-step solution

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

The cross product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is given by \(\mathbf{u}\times \mathbf{v}\), which can be denoted as \(\mathbf{w}\). Using the components of \( \mathbf{u} \) and \( \mathbf{v} \), \[\mathbf{w} = \langle (-10)(0)- (0)(6), -((-10)(0)-(-10)(7)), (7)(0) - (-10)(0)\rangle = \langle 0, 70, 0 \rangle\]Thus, the cross product of \( \mathbf{u} \) and \( \mathbf{v} \) is \(\mathbf{w}=\langle 0,70,0\rangle\).

Check if \( \mathbf{w} \) is orthogonal to \( \mathbf{u} \)

To check if two vectors are orthogonal, we calculate their dot product, which is zero if the vectors are orthogonal. The dot product of \( \mathbf{u} \) and \( \mathbf{w} \) is \(\mathbf{u} \cdot \mathbf{w}\), which gives\[\mathbf{u} \cdot \mathbf{w} = (-10)(0)+0+(6)(0) = 0\]Since the dot product of \( \mathbf{u} \) and \( \mathbf{w} \) equals 0, \( \mathbf{w} \) is orthogonal to \( \mathbf{u} \).

Check if \( \mathbf{w} \) is orthogonal to \( \mathbf{v} \)

Similarly, to check if \( \mathbf{w} \) is orthogonal to \( \mathbf{v} \), we calculate their dot product, which should also be zero. The dot product of \( \mathbf{v} \) and \( \mathbf{w} \) is \(\mathbf{v} \cdot \mathbf{w}\), which gives\[\mathbf{v} \cdot \mathbf{w} = (7)(0)+0+0 = 0\]Since the dot product of \( \mathbf{v} \) and \( \mathbf{w} \) equals 0, \( \mathbf{w} \) is orthogonal to \( \mathbf{v} \).

Key concepts

Vector Orthogonality

Understanding vector orthogonality is crucial in many fields such as physics, engineering, and computer graphics. Two vectors are considered orthogonal if they meet at a right angle, that is, a 90-degree angle. In a three-dimensional space, if you have two vectors, say \( \mathbf{a} \) and \( \mathbf{b} \), imagine them as arrows starting from the same point. If these arrows point in directions that are at right angles to each other, the vectors are orthogonal.

The concept can also be related to the more familiar idea of perpendicular lines in two-dimensional geometry. However, vector orthogonality doesn't solely depend on the spatial relationship; it is importantly characterized by their dot product. When two vectors are orthogonal, their dot product is zero. This is a defining property and provides a straightforward method to check the orthogonality of vectors without the need to visualize them in space.

In the given exercise, this concept is used to demonstrate that the cross product of two vectors results in a vector that is orthogonal to both original vectors.

Dot Product

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation combines the respective components from each vector and adds these together after multiplying.

To calculate the dot product of two vectors \( \mathbf{x} \) and \( \mathbf{y} \) with components \( x_1, x_2, \ldots \) and \( y_1, y_2, \ldots \), respectively, you multiply the corresponding components and sum those products: \[ \mathbf{x} \cdot \mathbf{y} = x_1 \cdot y_1 + x_2 \cdot y_2 + \ldots \]

If the result of this calculation is zero, as it is in the exercise provided, it indicates that the vectors are orthogonal to each other. This property is especially significant when discussing vector orthogonality because it offers a clear, quantitative way to determine the relationship between vectors in regards to their angle.

Vector Components

Every vector in a space can be broken down into components along the axes of that space. These components represent the influence of the vector along each axis. For a vector \( \mathbf{a} \) in three-dimensional space, it can be represented by its components along the x, y, and z axes: \( \mathbf{a} = \langle a_x, a_y, a_z \rangle \). Each component \( a_x, a_y \), and \( a_z \) corresponds to the projection of the vector along the respective axis.

In computations, working with vector components simplifies many operations, as seen in the calculation of both the cross product and the dot product. For the cross product, we calculate the components of the resulting vector by taking specific pairs of components from the initial vectors and applying specific subtraction and multiplication rules.

Understanding the vector components is foundational to working with both the cross product and the dot product, as the exercise shows. The components are effectively the building blocks of the vector, allowing us to perform these operations and help describe the vector's properties in relation to others.