91Ó°ÊÓ

Vectors \(\vec{A}\) and \(\vec{B}\) are in the \(x y\) -plane. Vector \(\vec{A}\) is in the \(+x\) - direction, and the direction of vector \(\overrightarrow{\boldsymbol{B}}\) is at an angle \(\theta\) from the \(+x\) -axis measured toward the \(+y\) -axis. (a) If \(\theta\) is in the range \(0^{\circ} \leq \theta \leq 180^{\circ}\), for what two values of \(\theta\) does the scalar product \(\vec{A} \cdot \vec{B}\) have its maximum magnitude? For each of these values of \(\theta,\) what is the magnitude of the vector product \(\vec{A} \times \vec{B} ?(b)\) If \(\theta\) is in the range \(0^{\circ} \leq \theta \leq 180^{\circ}\) for what value of \(\theta\) does the vector product \(\vec{A} \times \vec{B}\) have its maximum value? For this value of \(\theta,\) what is the magnitude of the scalar product \(\vec{A} \cdot \vec{B} ?(\mathrm{c})\) What is the angle \(\theta\) in the range \(0^{\circ} \leq \theta \leq 180^{\circ}\) for which \(\vec{A} \cdot \vec{B}\) is twice \(|\vec{A} \times \vec{B}| ?\)

Step by step solution

01

Maximum scalar product

For the maximum scalar product, we know that the dot product of two vectors \(\vec{A}\) and \(\vec{B}\) is given by \(\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos(\theta)\). The cosine function achieves its maximum value of 1 for two angles, 0 and 180 degrees, within the range 0 ≤ θ ≤ 180 degrees. Therefore, those are the two angles at which \(\vec{A} \cdot \vec{B}\) reaches its maximum magnitude.

02

Magnitude of vector product at those angles

Now, we need to compute the magnitude of the cross product for those angles. The cross product magnitude is given by \(|\vec{A}\times\vec{B}| = |\vec{A}||\vec{B}|\sin(\theta)\). For both theta=0 and theta=180 degrees, the sine function value is 0. Thus, \(|\vec{A}\times\vec{B}| = 0\) at those angles.

03

Maximum vector product

Next, we need to find the angle at which the cross product achieves its maximum value. The sine function achieves its maximum value of 1 for angle θ equals to 90 degrees within the range. Therefore, \(|\vec{A} \times \vec{B}|\) is maximized at θ = 90 degrees.

04

Scalar product at that angle

At θ=90 degrees, the cosine function value is 0. This means that \(\vec{A} \cdot \vec{B} = 0\) at θ=90 degrees.

05

Finding the angle for special condition

For the last part, we are looking for the angle at which the scalar product is twice the magnitude of the vector product, and so \(\vec{A} \cdot \vec{B} = 2|\vec{A}\times\vec{B}|\). Substituting dot and cross product representations, we obtain \(|\vec{A}||\vec{B}|\cos(\theta) = 2|\vec{A}||\vec{B}|\sin(\theta)\). Canceling out the magnitudes, we find that \(\frac{\cos(\theta)}{\sin(\theta)} = \frac{2\sin(\theta)}{\cos(\theta)}\), which simplifies to \(\tan(\theta) = 2\). Therefore, theta for which this condition is satisfied is approximately 63.4 degrees.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Dot Product

The dot product, also known as the scalar product, is a way to multiply two vectors and get a scalar (a regular number) as a result. When we calculate the dot product of two vectors \( \vec{A} \) and \( \vec{B} \), it tells us how much of one vector goes in the direction of the other. In mathematical terms, the dot product is defined as: \[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta) \] where \( \theta \) is the angle between the two vectors.

• If the angle \( \theta \) is 0 degrees, the dot product is maximum because \( \cos(0) = 1 \). This means the vectors are aligned in the same direction.
• If \( \theta \) is 180 degrees, the dot product is negative maximum because \( \cos(180) = -1 \). This indicates the vectors are in opposite directions.
• When \( \theta \) is 90 degrees, the dot product is zero, meaning the vectors are perpendicular.

Understanding the dot product is crucial because it reflects how vectors relate directionally to each other. It's helpful in numerous applications like determining the angle between two vectors or projecting one vector onto another.

Exploring the Cross Product

The cross product, also known as the vector product, differs from the dot product because it results in another vector, not a scalar. This new vector is perpendicular to the plane formed by the original vectors \( \vec{A} \) and \( \vec{B} \). The magnitude of the cross product can be calculated using the formula: \[ |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin(\theta) \] Here, \( \theta \) is again the angle between the two vectors.

• The cross product is maximized when \( \theta \) is 90 degrees because \( \sin(90) = 1 \). This is when the vectors are perpendicular to each other.
• For angles \( \theta = 0 \) or \( 180 \) degrees, the cross product is zero, indicating the vectors are either parallel or anti-parallel.

The direction of the resultant vector follows the right-hand rule. If you point your fingers in the direction of \( \vec{A} \) and curl them towards \( \vec{B} \), then your thumb points in the direction of \( \vec{A} \times \vec{B} \).Understanding the cross product is essential in physics, especially when dealing with rotational forces and moments.

Grasping Vector Angles

The angle between two vectors is a fundamental concept when studying vector operations, as it provides valuable insights into their orientation and relationship. This angle, denoted as \( \theta \), is pivotal in both the dot and cross product calculations.

• When vectors are aligned, \( \theta = 0 \) degrees, meaning they point in the same direction.
• A \( \theta = 90 \) degrees means the vectors are perpendicular and do not influence each other in terms of projection.
• A \( \theta = 180 \) degrees implies that the vectors are directly opposite.

The tangent of the angle \( \theta \) is often used to relate the dot and cross products, such as finding specific angles that satisfy given equations. For instance, when \( \tan(\theta) = 2 \), we solve for \( \theta \) to determine the angle relationship between the vectors based on specific criteria.By understanding vector angles, one can better interpret the dynamics between vectors, making this concept invaluable in fields such as engineering and computer graphics.