Problem 29 - Thomas Calculus

Question

Given nonzero vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w},\) use dot product and cross product notation, as appropriate, to describe the following. a. The vector projection of \(\mathbf{u}\) onto \(\mathbf{v}\) b. A vector orthogonal to \(\mathbf{u}\) and \(\mathbf{v}\) c. A vector orthogonal to \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{w}\) d. The volume of the parallelepiped determined by \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) e. A vector orthogonal to \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{u} \times \mathbf{w}\) f. A vector of length \(|\mathbf{u}|\) in the direction of \(\mathbf{v}\)

Step-by-Step Solution

Answer

a: projection formula, b: \(\mathbf{u} \times \mathbf{v}\), c: \((\mathbf{u} \times \mathbf{v}) \times \mathbf{w}\), d: volume formula, e: cross product, f: normalized \(\mathbf{v}\) times \(|\mathbf{u}|\).

Step 1: Vector Projection of \(\mathbf{u}\) onto \(\mathbf{v}\)

The vector projection of \(\mathbf{u}\) onto \(\mathbf{v}\) is given by the formula:\[\mathrm{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}\] This calculates a vector that represents how much of \(\mathbf{u}\) is in the direction of \(\mathbf{v}\).

Step 2: A Vector Orthogonal to \(\mathbf{u}\) and \(\mathbf{v}\)

A vector orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\) is the cross product \(\mathbf{u} \times \mathbf{v}\). The cross product of two vectors produces a vector that is perpendicular to the plane containing both vectors.

Step 3: A Vector Orthogonal to \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{w}\)

To find a vector orthogonal to both \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{w}\), we take the cross product of \(\mathbf{u} \times \mathbf{v}\) with \(\mathbf{w}\):\[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w}\]This will result in a vector orthogonal to these two vectors.

Step 4: Volume of the Parallelepiped

The volume of the parallelepiped determined by \(\mathbf{u}, \mathbf{v}, \text{and} \mathbf{w}\) is given by the scalar triple product:\[V = |\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})|\] This expression calculates the absolute value of the dot product of \(\mathbf{u}\) with the cross product of \(\mathbf{v}\) and \(\mathbf{w}\).

Step 5: A Vector Orthogonal to \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{u} \times \mathbf{w}\)

To find a vector orthogonal to both \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{u} \times \mathbf{w}\), take their cross product:\[(\mathbf{u} \times \mathbf{v}) \times (\mathbf{u} \times \mathbf{w})\] This gives a vector perpendicular to both input cross products.

Step 6: Vector of Length \(|\mathbf{u}|\) in the Direction of \(\mathbf{v}\)

To create a vector with the magnitude of \(|\mathbf{u}|\) in the direction of \(\mathbf{v}\), you normalize \(\mathbf{v}\) and multiply it by \(|\mathbf{u}|\):\[\frac{\mathbf{v}}{|\mathbf{v}|} \times |\mathbf{u}|\]This results in a vector pointing in the direction of \(\mathbf{v}\) with the desired magnitude.

Key Concepts

Vector Projection

Vector projection is a way of translating one vector onto another, showing how much of one vector lies in the direction of the other. In mathematical terms, the projection of a vector \( \mathbf{u} \) onto a vector \( \mathbf{v} \) is given by the formula: \( \mathrm{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v} \).
This equation involves the dot product and tells us how much of \( \mathbf{u} \) is aligned with \( \mathbf{v} \).
To better understand this, picture

• \( \mathbf{u} \) as a shadow projected onto \( \mathbf{v} \)
• The resulting vector points in the same direction as \( \mathbf{v} \)
• This projection is essentially the amount of \( \mathbf{u} \) that goes in the direction of \( \mathbf{v} \).

The magnitude of this projection represents the actual distance along \( \mathbf{v} \) that \( \mathbf{u} \) occupies.

Cross Product

The cross product is a vector operation that finds a vector perpendicular to two given vectors. This operation is only applicable in three-dimensional space, where it creates a vector orthogonal to the plane defined by the original pair of vectors.
The cross product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is denoted as \( \mathbf{u} \times \mathbf{v} \). Mathematically, the cross product can be determined using the formula:

• \( \mathbf{u} \times \mathbf{v} = (u_2v_3 - u_3v_2)\mathbf{i} + (u_3v_1 - u_1v_3)\mathbf{j} + (u_1v_2 - u_2v_1)\mathbf{k} \)

Here's why the cross product is useful:

• The result is a new vector that is at a right angle (orthogonal) to the original vectors
• The magnitude of this vector is proportional to the area of the parallelogram formed by the initial vectors \( \mathbf{u} \) and \( \mathbf{v} \).
• This can be utilized to find axes for rotation, or to detect parallel planes or surfaces.

Scalar Triple Product

The scalar triple product is a vital concept for computing volumes in vector calculus. Specifically, it calculates the volume of a parallelepiped — a three-dimensional figure formed by three vectors. The scalar triple product of vectors \( \mathbf{u}, \mathbf{v} \) and \( \mathbf{w} \) is expressed by \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \), where:

• First, the cross product \( \mathbf{v} \times \mathbf{w} \) forms a vector perpendicular to both \( \mathbf{v} \) and \( \mathbf{w} \)
• Second, the dot product measures how much the third vector \( \mathbf{u} \) aligns with this resulting perpendicular vector

The absolute value of this scalar quantity gives the volume of the parallelepiped.
It provides an easy way to determine the total space enclosed by these vectors, an essential task in physics and engineering applications.

Dot Product

The dot product is a fundamental vector operation that finds the scalar product of two vectors. It measures how much one vector extends in the direction of another. If we have two vectors \( \mathbf{u} \) and \( \mathbf{v} \), the dot product \( \mathbf{u} \cdot \mathbf{v} \) is calculated as:

• \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \)

This operation is critical for several reasons:

• It results in a scalar, not a vector
• The dot product is zero if the vectors are orthogonal (perpendicular)
• The value of the dot product relates to the angle between the vectors, providing insight into their relative orientation

The dot product makes it possible to decompose vectors, compute angles, and determine orthogonality, making it crucial in spatial calculations and projections.