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Chapter 2: Problem 274

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

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

Step by step solution

01

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.
02

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.
03

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.
04

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.
05

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=\(\frac{π}{2}\).

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