Question

\(\mathrm{A}^{\rightarrow} \times \mathrm{B}^{\rightarrow}=\mathrm{C}^{-}\), Then \(\mathrm{C}^{\rightarrow}\) is perpendicular to (A) \(\mathrm{A}^{\rightarrow}\) only (B) \(\mathrm{B}^{\rightarrow}\) only (C) \(\mathrm{A}^{\rightarrow}\) and \(\mathrm{B}^{\rightarrow}\) both when the angle between them is \(\ldots\) (D) \(\mathrm{A}^{-}\) and \(\mathrm{B}^{\rightarrow}\) both whatever to be the angle between them

Short answer

The correct answer is (D): \(C^{\rightarrow}\) is perpendicular to both \(\mathrm{A}^{-}\) and \(\mathrm{B}^{\rightarrow}\), irrespective of the angle between them, because it is a result of their cross product.

Step-by-step solution

Recognize the properties of the cross product

The cross product of two vectors A and B, denoted as \(A \times B\), results in a new vector C, which has the following properties: 1. The magnitude of C is equal to the product of the magnitudes of A and B and the sine of the angle between them: \(|C| = |A||B|\sin(\theta)\). 2. Vector C is perpendicular to both A and B. So the statement states that the cross product of A and B results in vector C, and C is negative.

Analyze the options

Here are the options presented in the question: (A) C is perpendicular to A only. (B) C is perpendicular to B only. (C) C is perpendicular to both A and B when the angle between them is ... (D) C is perpendicular to A and B whatever the angle between them. Since the cross product of two vectors is a vector that is perpendicular to both A and B, the only option that matches the properties of the cross product is option (D).

Choose the correct answer and explain

The correct answer is (D): \(C^{\rightarrow}\) is perpendicular to both \(\mathrm{A}^{-}\) and \(\mathrm{B}^{\rightarrow}\), irrespective of the angle between them, because it is a result of their cross product. The cross product of any two vectors always produces a vector that is perpendicular to the original vectors, regardless of the angle between the original vectors. Note that the statement does not impact the final answer since only the direction is negative, not the property of being perpendicular.

Key concepts

Vector Properties

Understanding vector properties is essential when working with cross products. Vectors have both a magnitude (size) and a direction, which makes them different from simple scalar quantities like mass or temperature. When you multiply two vectors using the cross product, you end up with another vector. This resulting vector exhibits specific properties that are important to understand:

• The direction of the resultant vector, in the case of a cross product, is determined using the right-hand rule. This rule helps figure out the orientation of the vector in three-dimensional space.
• The magnitude of the cross product vector is given by the product of the magnitudes of the original vectors and the sine of the angle between them, expressed mathematically as: \[ |\mathbf{C}| = |\mathbf{A}||\mathbf{B}|\sin(\theta) \]
• The cross product is always perpendicular to the plane defined by the original two vectors, which is crucial for determining its direction.

Grasping these properties helps in visualizing how vectors interact in space, which is fundamental for vector algebra.

Perpendicular Vectors

In relation to the cross product, one of the fascinating properties is that the resultant vector is always perpendicular to the original vectors. This means that the resulting vector \( \mathbf{C} \) from the cross product of \( \mathbf{A} \) and \( \mathbf{B} \) can be imagined as a vector shooting out at right angles from the plane they form.

• Understanding the concept of perpendicular vectors can make visualizing problems involving torque, rotational motion, or even electromagnetic fields much simpler.
• In three-dimensional space, two vectors are perpendicular if their dot product is zero. Meanwhile, the cross product ensures perpendicularity by default when it produces a resulting vector.

This inherent perpendicular nature of the cross product vectors is what makes them so practical for real-world applications, especially in physics and engineering.

Magnitude and Direction

Magnitude and direction are crucial elements of understanding vectors and their products. For a cross product, the magnitude is a measure of how strong or significant the resultant vector is relative to the original vectors.

• The magnitude of a cross product not only takes into account the magnitudes of the original vectors \( \mathbf{A} \) and \( \mathbf{B} \), but it also crucially depends on the sine of the angle \( \theta \) between them. This mechanic ensures that the resulting vector vanishes if the two vectors are parallel or become maximum if they are perpendicular.
• Direction, based on the right-hand rule, shows the orientation of the resulting vector, which is perpendicular to the original vectors. You curl your right hand's fingers from \( \mathbf{A} \) towards \( \mathbf{B} \) and your thumb points \( \mathbf{C} \) direction.

Without a clear understanding of these aspects, interpreting vectors and their interactions in different space orientations would be much harder.

Angle Between Vectors

The angle between vectors plays a pivotal role in calculating the magnitude of a cross product. This angle is the factor that determines how "spread out" the vectors are from one another in space.

• When the angle \( \theta \) is 90 degrees (or \( \frac{\pi}{2} \) radians), the sine of the angle becomes 1, thereby maximizing the magnitude of the resultant vector \( \mathbf{C} \). This condition means that the vectors are perpendicular, and thus the cross product will fully represent area spanned by the vectors.
• If the angle is 0 or 180 degrees, where vectors are collinear, the sine of these angles is zero. Hence, the resultant cross product is a zero vector, highlighting the lack of any area between the vectors.

Thus, comprehending the angle between vectors is crucial for predicting the behavior of vectors under multiplication.