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Chapter 1: Problem 36

Three vectors satisfy the relation \(\vec{A} \cdot \vec{B}=0\) and \(\vec{A} \cdot \vec{C}=0\), then \(\vec{A}\) is parallel to (A) \(\vec{C}\) (B) \(\vec{B}\) (C) \(\vec{B} \times \vec{C}\) (D) \(\vec{B} \cdot \vec{C}\)

Short Answer

Expert verified
The correct option is (C) \(\vec{A}\) is parallel to \(\vec{B} \times \vec{C}\), since \(\vec{A}\) is orthogonal to both \(\vec{B}\) and \(\vec{C}\), it is parallel to their cross product.

Step by step solution

01

Review the dot product properties

Recall the definition of the dot product of two vectors: \(\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta)\) where \(\theta\) is the angle between the two vectors, and \(|\vec{A}|\) and \(|\vec{B}|\) are the magnitudes of the respective vectors. One important property of the dot product is that: If \(\vec{A} \cdot \vec{B} = 0\), then either \(\vec{A}\) or \(\vec{B}\) is a zero vector, or \(\theta = 90^{\circ}\), which means the two vectors are orthogonal.
02

Relate given conditions to parallel vectors

Since \(\vec{A} \cdot \vec{B} = 0\) and \(\vec{A} \cdot \vec{C} = 0\), we know that vectors \(\vec{A}\) and \(\vec{B}\), and \(\vec{A}\) and \(\vec{C}\), are orthogonal to each other. Now, let's recall the properties of the cross product. Given two vectors \(\vec{P}\) and \(\vec{Q}\), their cross product \(\vec{P} \times \vec{Q}\) is a vector that is orthogonal to both \(\vec{P}\) and \(\vec{Q}\).
03

Determine the correct option

Since \(\vec{A}\) is orthogonal to both \(\vec{B}\) and \(\vec{C}\), it means that it is parallel to the vector that is formed by the cross product of \(\vec{B}\) and \(\vec{C}\). Therefore, the correct option is: (C) \(\vec{A}\) is parallel to \(\vec{B} \times \vec{C}\).

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