Question

If \(\mathrm{A}^{\rightarrow} \cdot \mathrm{B}^{\rightarrow}=0\) then (A) \(\left|\mathrm{A}^{\rightarrow}\right|\) must be zero (B) \(\mathrm{B}^{\rightarrow} \mid\) must be zero (C) either \(\mathrm{A}^{\rightarrow}=0, \mathrm{~B}^{\rightarrow}=0\) or \(\theta=0\) (D) either \(\mathrm{A}^{\rightarrow}=0, \mathrm{~B}^{\rightarrow}=0\) or \(\theta=(\pi / 2)\)

Short answer

The correct answer is (D) either 𝐀=0, 𝐁=0, or θ=\(\frac{π}{2}\).

Step-by-step solution

Definition of Dot Product

The dot product of two vectors, 𝐀 and 𝐁, is given by: \(𝐀⋅𝐁 = |𝐀||𝐁| \cos{θ}\), where |𝐀| and |𝐁| are the magnitudes of 𝐀 and 𝐁, respectively, and θ is the angle between them.

Analyzing Option (A)

According to option (A), if 𝐀⋅𝐁=0, then |𝐀|=0. However, since θ can be \(\frac{π}{2}\) such that the cosine becomes 0, |𝐀| does not have to be zero. So, option (A) is not necessarily true.

Analyzing Option (B)

According to option (B), if 𝐀⋅𝐁=0, then |𝐁|=0. Similar to the reasoning behind option (A), θ can also be \(\frac{π}{2}\), so |𝐁| does not have to be zero. Therefore, option (B) is not necessarily true.

Analyzing Option (C)

Option (C) states that if 𝐀⋅𝐁=0, then either 𝐀=0, 𝐁=0, or θ=0. Using the formula \(𝐀⋅𝐁 = |𝐀||𝐁| \cos{θ}\), we see that this option is not necessarily true. Since if θ=0, then the cosine of θ is 1, and 𝐀⋅𝐁 might not be zero.

Analyzing Option (D)

Option (D) states that if 𝐀⋅𝐁=0, then either 𝐀=0, 𝐁=0, or θ=\(\frac{π}{2}\). Using the formula \(𝐀⋅𝐁 = |𝐀||𝐁| \cos{θ}\), we can see that if either |𝐀|=0 or |𝐁|=0, the dot product will be zero. Moreover, if θ=\(\frac{π}{2}\), the cosine of θ is 0, so the dot product will also be zero. Thus, option (D) is necessarily true. So, the correct answer is (D) either 𝐀=0, 𝐁=0, or θ=\(\frac{π}{2}\).

Key concepts

Vector Magnitude

Vectors have both magnitude and direction. The magnitude of a vector is the length or size of the vector, and it is a non-negative value. For example, if we have a vector \( \mathbf{A} = (a_1, a_2, a_3) \), the magnitude is given by the formula: \[ |\mathbf{A}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \] This formula can be applied to any vector, whether it's in two or three dimensions.

• The magnitude tells us how "strong" the vector is in terms of its length.
• It is always positive or zero, never negative.

Understanding magnitude is essential in comparing the lengths of vectors and in operations like normalizing a vector, which means making it a unit vector.

Angle Between Vectors

The angle between two vectors, often denoted as \( \theta \), is crucial for understanding their relative orientation. Given two vectors \( \mathbf{A} \) and \( \mathbf{B} \), the angle between them can be found using the dot product formula. The dot product is expressed as: \[ \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}| \cos{\theta} \] Here, \( \cos{\theta} \) represents the cosine of the angle between the two vectors. If the dot product is zero, then either one or both vectors have zero magnitude, or the angle \( \theta \) is \( \frac{\pi}{2} \) radians (90 degrees).

• When the angle is 0, the vectors are in the same direction.
• When the angle is \( \frac{\pi}{2} \), the vectors are perpendicular (orthogonal).
• Understanding the angle helps in determining how vectors interact with each other in space.

Orthogonal Vectors

Orthogonal vectors are vectors that are at right angles (90 degrees) to each other. In terms of vector mathematics, when two vectors are orthogonal, their dot product is zero. This is because the cosine of 90 degrees is zero:

• Mathematically, \( \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}| \cos{\theta} = 0 \). This implies \( \cos{\theta} = 0 \).
• This zero result signifies no projection of one vector on the other.

Orthogonality is a key concept in various mathematical and engineering applications, such as finding perpendicular directions or indicating independence between quantities. Recognizing orthogonal vectors is useful in simplifying vector problems and verifying if vectors form a basis.

Vector Mathematics

Vector mathematics involves various operations that fully describe the properties and interactions of vectors. The dot product is a fundamental operation that returns a scalar by multiplying corresponding components of two vectors and summing the results:

• The dot product is useful for determining angles and checking orthogonality as seen before.
• Vectors can also be added or subtracted to find resultant vectors, which is essential in physics and engineering for understanding forces and velocities.

Besides the dot product, another critical operation is the cross product, which produces a vector perpendicular to the two original vectors. These operations form the backbone of vector mathematics, providing tools to model and solve problems in multidimensional spaces.