Problem 9 - Calculus: Early Transcendentals

Question

Prove that for all vectors \(\boldsymbol{u}\) and \(\boldsymbol{v},(\boldsymbol{u} \times \boldsymbol{v}) \cdot \boldsymbol{v}=0 .\)

Step-by-Step Solution

Answer

The cross product is perpendicular to its operands, ensuring a zero dot product.

Step 1: Understand the Cross Product

The cross product \( \boldsymbol{u} \times \boldsymbol{v} \) results in a vector that is perpendicular to both \( \boldsymbol{u} \) and \( \boldsymbol{v} \). This means that the resultant vector is orthogonal to the original vectors.

Step 2: Recall the Dot Product Property

The dot product of two vectors is zero if and only if the two vectors are perpendicular to each other. Thus, if a vector \( \boldsymbol{w} \) is perpendicular to \( \boldsymbol{v} \), then \( \boldsymbol{w} \cdot \boldsymbol{v} = 0 \).

Step 3: Apply the Cross and Dot Products

Since \( \boldsymbol{u} \times \boldsymbol{v} \) is perpendicular to \( \boldsymbol{v} \), applying the dot product, we have \( (\boldsymbol{u} \times \boldsymbol{v}) \cdot \boldsymbol{v} = 0 \).

Step 4: Conclusion

Therefore, for any vectors \( \boldsymbol{u} \) and \( \boldsymbol{v} \), the statement \( (\boldsymbol{u} \times \boldsymbol{v}) \cdot \boldsymbol{v} = 0 \) holds true, as the cross product is guaranteed to be orthogonal to both original vectors.

Key Concepts

Cross Product

The cross product is a fascinating operation from vector calculus that combines two vectors to produce a new vector. This new vector holds a unique property: it's perpendicular to both of the original vectors involved in the operation.
To create the cross product between two vectors, say \( \boldsymbol{u} \) and \( \boldsymbol{v} \), we use the notation \( \boldsymbol{u} \times \boldsymbol{v} \). The resultant vector is often used in physics and engineering, especially when dealing with rotational forces or torques.

• Magnitude: The magnitude of the cross product \( \| \boldsymbol{u} \times \boldsymbol{v} \| \) equals the area of the parallelogram formed by \( \boldsymbol{u} \) and \( \boldsymbol{v} \).
• Direction: Determined by the right-hand rule, where your thumb points in the direction of the resultant vector when your fingers curl from \( \boldsymbol{u} \) to \( \boldsymbol{v} \).

Cross products are zero if the vectors are parallel since the angle between them, in this case, is either \( 0 \) or \( 180 \) degrees, leading to no defined plane to form a vector that is perpendicular.

Dot Product

The dot product is another crucial operation in vector calculus, but unlike the cross product, it results in a scalar (a number without direction). The dot product between vectors \( \boldsymbol{u} \) and \( \boldsymbol{v} \) is represented as \( \boldsymbol{u} \cdot \boldsymbol{v} \).The formula for the dot product is as follows:\[\boldsymbol{u} \cdot \boldsymbol{v} = \| \boldsymbol{u} \| \| \boldsymbol{v} \| \cos(\theta)\]where \( \theta \) is the angle between the two vectors.

• Interpreting Zero: If \( \boldsymbol{u} \cdot \boldsymbol{v} = 0 \), it indicates that the vectors are orthogonal, or perpendicular to each other.
• Applications: The dot product helps in projecting one vector onto another, calculating work done by a force, or finding angles between vectors.

For our exercise, the product between the cross product \( \boldsymbol{u} \times \boldsymbol{v} \) and vector \( \boldsymbol{v} \) equals zero, showing that they are orthogonal.

Orthogonal Vectors

Two vectors are said to be orthogonal if they meet at a right angle. In simpler terms, their directions don't share any component visually resembling a 'T'.

• Mathematics of Orthogonality: Orthogonal vectors have a pivotal characteristic, which is that their dot product equals zero. This is because the angle \( \theta \) between them is \( 90 \) degrees, leading to \( \cos(\theta) = 0 \).
• Geometrical interpretation: If you imagine each vector as a line pouring out from a common origin, orthogonal vectors extend so that they form an 'L' shape.
• Real-life Importance: Orthogonality is crucial in many fields such as computer graphics for light reflection, signal processing for noise reduction, and more.

In the exercise, the vector derived from \( \boldsymbol{u} \times \boldsymbol{v} \) is orthogonal to both \( \boldsymbol{u} \) and \( \boldsymbol{v} \), thus confirming the statement \( (\boldsymbol{u} \times \boldsymbol{v}) \cdot \boldsymbol{v} = 0 \). This confirmation reinforces the beautiful symmetry and consistency within the rules of vector calculus.