Problem 29 - University Calculus: Early Transcendentals

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. proj_v u = (u·v/v·v)v, b. u × v, c. (u × v) × w, d. |u · (v × w)|, e. (u × v) × (u × w), f. |u| (v/|v|).

Step 1: Vector Projection of u onto v

The vector projection of \( \mathbf{u} \) onto \( \mathbf{v} \) is given by the formula: \[ \text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v} \]This formula represents the shadow of vector \( \mathbf{u} \) onto the line defined by \( \mathbf{v} \). The expression \( \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \) is a scalar that scales vector \( \mathbf{v} \).

Step 2: Vector Orthogonal to u and v

A vector orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \) is given by their cross product: \[ \mathbf{u} \times \mathbf{v} \]This vector is perpendicular to the plane formed by \( \mathbf{u} \) and \( \mathbf{v} \). The cross product results in a vector that is orthogonal to both of the original vectors involved.

Step 3: Vector Orthogonal to (u × v) and w

A vector orthogonal to \( \mathbf{u} \times \mathbf{v} \) and \( \mathbf{w} \) can be found using the cross product of these two vectors:\[ (\mathbf{u} \times \mathbf{v}) \times \mathbf{w} \]The result is a vector perpendicular to both \( \mathbf{u} \times \mathbf{v} \) and \( \mathbf{w} \). The cross product provides a simple way to determine a vector orthogonal to two given vectors.

Step 4: Volume of the Parallelepiped

The volume of the parallelepiped determined by the vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w} \) is given by the scalar triple product:\[ |\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})| \]This calculation involves the determinant of a 3x3 matrix formed by the components of the vectors \( \mathbf{u}, \mathbf{v}, \text{ and } \mathbf{w} \). The absolute value gives the volume.

Step 5: Vector Orthogonal to (u × v) and (u × w)

A vector orthogonal to both \( \mathbf{u} \times \mathbf{v} \) and \( \mathbf{u} \times \mathbf{w} \) is given by their cross product:\[ (\mathbf{u} \times \mathbf{v}) \times (\mathbf{u} \times \mathbf{w}) \]This result is orthogonal to the plane defined by each pair of crossed vectors.

Step 6: Vector of Length |u| in the Direction of v

To find a vector of length \(|\mathbf{u}|\) in the direction of \(\mathbf{v}\), first find the unit vector in the direction of \(\mathbf{v}\):\[ \hat{\mathbf{v}} = \frac{\mathbf{v}}{\| \mathbf{v} \|} \]Then scale this unit vector by the magnitude of \(\mathbf{u}\):\[ \mathbf{a} = |\mathbf{u}| \cdot \hat{\mathbf{v}} = |\mathbf{u}| \cdot \frac{\mathbf{v}}{\| \mathbf{v} \|} \]This results in a vector \( \mathbf{a} \) that has the specified magnitude and direction.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a vital operation in vector algebra. It combines two vectors to produce a scalar.
The formula for the dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is:\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\]where \( a_i \) and \( b_i \) are the components of vectors \( \mathbf{a} \) and \( \mathbf{b} \), respectively.

• Magnitude and Angle: The dot product is related to the magnitudes of the vectors and the cosine of the angle between them: \( \mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos \theta \).
• Orthogonality: Two vectors are orthogonal (perpendicular) if their dot product is zero.
• Applications: Used in projections, finding angles, and determining orthogonality.

When working with projections, the dot product is essential for calculating the scalar portion of the projection formula. This scalar defines how much of one vector lies along another. This insight is quite powerful in physics for analyzing forces, work, and more.

Cross Product

The cross product, distinct from the dot product, results in a vector rather than a scalar. It is denoted by \( \mathbf{a} \times \mathbf{b} \) and produces a vector perpendicular to the plane containing \( \mathbf{a} \) and \( \mathbf{b} \).
The formula for the cross product is:\[\mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)\]This results in a new vector, and

• Magnitude: The magnitude of the cross product is \( \|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \|\mathbf{b}\| \sin \theta \), which represents the area of the parallelogram spanned by vectors \( \mathbf{a} \) and \( \mathbf{b} \).
• Direction: The direction is determined by the right-hand rule: point your index finger in the direction of \( \mathbf{a} \), your middle finger in the direction of \( \mathbf{b} \), and your thumb will point in the direction of \( \mathbf{a} \times \mathbf{b} \).
• Orthogonality: Resulting vector is orthogonal to both \( \mathbf{a} \) and \( \mathbf{b} \).

The cross product is widely used in physics, such as in torque calculations and generating normal vectors to surfaces.

Projection of a Vector

Projecting one vector onto another is like casting a shadow of one vector onto the line defined by the other. The projection of vector \( \mathbf{u} \) onto vector \( \mathbf{v} \) is represented as:\[\text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}\]This formula calculates a vector that lies along the direction of \( \mathbf{v} \), scaling it appropriately.
Some key points:

• Scalar Factor: \( \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \) is the scalar that represents how much of \( \mathbf{u} \) is in the direction of \( \mathbf{v} \).
• Length: The length of this projection is \( \left| \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \|\mathbf{v}\| \right| \).
• Uses: Essential in physics and engineering to resolve forces and other vectors into simpler components.

This concept is crucial for analyzing and simplifying vector relationships in both mathematical and real-world contexts.

Triple Scalar Product

The triple scalar product involves combining three vectors, \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \), using both the dot and cross products.
This operation is useful for finding the volume of a parallelepiped (a 3D figure with parallelogram faces). The volume is given by the absolute value of the triple scalar product:\[| \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) |\]Here's how it works:

• Order Matters: The cross product creates a vector, and the dot product with a third vector results in a scalar.
• Volume Indicator: The absolute value implies the non-negative quantity—regardless of orientation.
• Jacobian Determination: In vector calculus, the determinant of a 3x3 matrix consisting of these vectors contributes to transformations.

The triple scalar product helps us explore volume and orientation in three-dimensional space, vital for fields ranging from geometry to physics.