Problem 56 - University Physics with Modern Physics

Question

By making simple sketches of the appropriate vector products, show that \((a) \vec{A} \cdot \vec{B}\) can be interpreted as the product of the magnitude of \(\overrightarrow{\boldsymbol{A}}\) times the component of \(\overrightarrow{\boldsymbol{B}}\) along \(\overrightarrow{\boldsymbol{A}}\), or the magnitude of \(\vec{B}\) times the component of \(\vec{A}\) along \(\overrightarrow{\boldsymbol{B}}\) (b) \(|\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}|\) can be interpreted as the product of the magnitude of \(\overrightarrow{\boldsymbol{A}}\) times the component of \(\overrightarrow{\boldsymbol{B}}\) perpendicular to \(\overrightarrow{\boldsymbol{A}},\) or the magnitude of \(\overrightarrow{\boldsymbol{B}}\) times the component \(\overrightarrow{\boldsymbol{A}}\) perpendicular to \(\overrightarrow{\boldsymbol{B}}\).

Step-by-Step Solution

Answer

(a) \( \vec{A} \cdot \vec{B} \) can be seen as projected magnitudes; (b) \( |\vec{A} \times \vec{B}| \) as perpendicular magnitudes.

Step 1: Understanding Dot Product

The dot product \( \vec{A} \cdot \vec{B} \) is defined as the product of the magnitudes of \( \vec{A} \) and \( \vec{B} \) with the cosine of the angle \( \theta \) between them: \( \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta) \). This measures how much of one vector extends in the direction of another.

Step 2: Interpreting Dot Product as a Projection

The dot product \( \vec{A} \cdot \vec{B} \) can be seen as the product of \( |\vec{A}| \) and the component of \( \vec{B} \) in the direction of \( \vec{A} \), which is \( |\vec{B}| \cos(\theta) \). This can be interpreted with \( \vec{B}_{\text{parallel}} = |\vec{B}| \cos(\theta) \hat{A} \) where \( \hat{A} \) is the unit vector in the direction of \( \vec{A} \).

Step 3: Alternate Interpretation of Dot Product

Similarly, \( \vec{A} \cdot \vec{B} \) can be interpreted as the magnitude of \( |\vec{B}| \cos(\theta) \) multiplied by \( |\vec{A}| \), where \( \vec{A} \) is projected onto \( \vec{B} \). This reflects the symmetry in projections between the two vectors.

Step 4: Understanding Cross Product Magnitude

The magnitude of the cross product \( |\vec{A} \times \vec{B}| \) is given by \( |\vec{A}||\vec{B}| \sin(\theta) \). This represents the area of the parallelogram spanned by \( \vec{A} \) and \( \vec{B} \).

Step 5: Interpreting Cross Product for Perpendicular Components

\( |\vec{A} \times \vec{B}| \) can be described as the magnitude of \( |\vec{A}| \) times the component of \( \vec{B} \) that is perpendicular to \( \vec{A} \), so \( \vec{B}_{\perp} \) is \( |\vec{B}| \sin(\theta) \).

Step 6: Alternate Interpretation of Cross Product

Conversely, \( |\vec{A} \times \vec{B}| \) is equivalent to the product of the magnitude of \( |\vec{B}| \) and the perpendicular component of \( \vec{A} \), where \( \vec{A}_{\perp} = |\vec{A}| \sin(\theta) \).

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics that helps measure how much one vector "stretches" in the direction of another. Imagine you have two vectors, \( \vec{A} \) and \( \vec{B} \), at an angle \( \theta \) to one another. The dot product, represented as \( \vec{A} \cdot \vec{B} \), is calculated using the formula:\[ \vec{A} \cdot \vec{B} = |\vec{A}|\, |\vec{B}| \cos(\theta) \]This formula indicates that to compute the dot product, you multiply the magnitude (length) of both vectors and then multiply this by the cosine of the angle between them. Breaking it down further:

• The magnitude \( |\vec{A}| |\vec{B}| \) gives you a basic idea of how "big" the vectors are.
• The cosine factor \( \cos(\theta) \) captures the extent of alignment between \( \vec{A} \) and \( \vec{B} \). If they point in the same or opposite directions, the cosine is close to 1 or -1; if they are perpendicular, it's zero.

This operation can be perceived as the projection of one vector onto another. For instance, if you want to know how much of \( \vec{B} \) is pointing in the direction of \( \vec{A} \), you look at the component of \( \vec{B} \) along \( \vec{A} \), which is \( |\vec{B}| \cos(\theta) \). Thus, a large dot product implies that the two vectors are aligned closely.

Cross Product

The cross product is another essential vector operation, which finds a vector perpendicular to the plane formed by two given vectors. Unlike the dot product, the cross product results in a vector rather than a scalar. To calculate the magnitude of the cross product between vectors \( \vec{A} \) and \( \vec{B} \), use the formula:\[ |\vec{A} \times \vec{B}| = |\vec{A}|\, |\vec{B}| \sin(\theta) \]In this formula:

• \( |\vec{A}| \) and \( |\vec{B}| \) represent the magnitudes of the respective vectors.
• \( \sin(\theta) \) measures how perpendicular the vectors are to each other, reaching its maximum when they are perpendicular.

The magnitude obtained represents the area of a parallelogram that the two vectors span. The direction follows the right-hand rule - if you point your right-hand fingers from \( \vec{A} \) to \( \vec{B} \), your thumb points in the direction of \( \vec{A} \times \vec{B} \).
Understanding this concept is crucial, particularly in physics, where the cross product helps compute torque and angular momentum.

Vector Magnitude

Vector magnitude is a core concept in vector analysis that refers to the "size" or "length" of a vector. If you consider a vector \( \vec{A} = (a_1, a_2, a_3) \) in three-dimensional space, its magnitude is calculated using the Pythagorean theorem:\[ |\vec{A}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \]This formula essentially extends the Pythagorean theorem to three dimensions and can also be applied in any number of dimensions by adding more components \( a_n \). Knowing the magnitude of a vector is essential when operating with vectors because:

• It allows you to assess the "strength" or "force" a vector can exert when it represents physical concepts like velocity or force.
• It's used in normalizing vectors, which means converting a vector so that it maintains its direction but has a unit length of 1.

By understanding the magnitude of vectors, you're able to better interpret the effects and applications of vectors in real-world scenarios. This understanding will also aid in more complex operations like unit vector calculations and normalization.