Question

If \(\mathbf{A}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k}\) and \(\mathbf{A} \cdot \mathbf{B}=0,\) does it follow that \(\mathbf{B}=0 ?(\) Either prove that it does or give a specific example to show that it doesn't.) Answer the same question if \(\mathbf{A} \times \mathbf{B}=\mathbf{0} .\) And again answer the same question if \(\mathbf{A} \cdot \mathbf{B}=0\) and \(\mathbf{A} \times \mathbf{B}=\mathbf{0}\).

Short answer

\(\textbf{B}\) does not have to be zero if \(\textbf{A} \cdot \textbf{B} = 0\), may have to be zero or a multiple of \(\textbf{A}\) if \(\textbf{A} \times \textbf{B} = \textbf{0}\), and must be zero if both conditions are satisfied.

Step-by-step solution

Understand the given vector for the initial conditions

The vector \(\textbf{A} = 2\textbf{i} - 3\textbf{j} + \textbf{k}\) is given. The dot product and cross product conditions involving vector \(\textbf{A}\) have to be analyzed.

Analyze the condition \(\textbf{A} \cdot \textbf{B} = 0\)

Recall that the dot product of two vectors is zero if and only if the vectors are perpendicular. Hence, if \(\textbf{A} \cdot \textbf{B} = 0\), it means \(\textbf{B}\) is perpendicular to \(\textbf{A}\). However, this does not necessarily mean that \(\textbf{B} = 0\). For instance, \(\textbf{B} = \textbf{i} + 3\textbf{j} - 2\textbf{k}\) is perpendicular to \(\textbf{A}\) but \(\textbf{B} \eq 0\). Hence, \(\textbf{B}\) does not have to be zero.

Analyze the condition \(\textbf{A} \times \textbf{B} = \textbf{0}\)

Recall that the cross product of two vectors is zero if and only if the vectors are parallel or one of them is zero. Thus, if \(\textbf{A} \times \textbf{B} = \textbf{0}\), it implies that \(\textbf{B}\) is either a multiple of \(\textbf{A}\) or \(\textbf{B} = 0\). For instance, \(\textbf{B} = k\textbf{A}\) where \(\textbf{k}\) is a scalar will fulfill this condition.

Analyze the condition \(\textbf{A} \cdot \textbf{B} = 0\) and \(\textbf{A} \times \textbf{B} = \textbf{0}\)

When both \(\textbf{A} \cdot \textbf{B} = 0\) and \(\textbf{A} \times \textbf{B} = \textbf{0}\) are satisfied, \(\textbf{B}\) must simultaneously be perpendicular to \(\textbf{A}\) and parallel to \(\textbf{A}\). The only vector that satisfies both conditions is the zero vector. Hence, \(\textbf{B}\ must be \textbf{0}\).

Key concepts

Dot Product

The dot product is a fundamental operation in vector analysis. It is defined as the product of the magnitudes of two vectors and the cosine of the angle between them. Mathematically, the dot product of vectors \(\textbf{A}\textbf{B} \) can be written as: \[ \textbf{A} \cdot \textbf{B} = |\textbf{A}| |\textbf{B}| \cos(\theta).\] In component form, if \( \textbf{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \) and \( \textbf{B} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k},\) then their dot product is: \[ \textbf{A} \cdot \textbf{B} = a_1 b_1 + a_2 b_2 + a_3 b_3 \] An important property is that if \( \textbf{A} \cdot \textbf{B} = 0,\) the vectors are perpendicular. However, this doesn't imply that one of the vectors is a zero vector, they could be different non-zero vectors.

Cross Product

The cross product is another crucial operation in vector analysis, mainly used in three dimensions. The cross product of two vectors \( \textbf{A} \textbf{B} \) results in a vector perpendicular to both \( \textbf{A} \) and \( \textbf{B}.\) The magnitude of the cross product is given by the area of the parallelogram formed by \( \textbf{A} \) and \( \textbf{B}.\) Mathematically, the cross product is: \[ \textbf{A} \times \textbf{B} = |\textbf{A}| |\textbf{B}| \sin(\theta) \hat{n},\] where \( \hat{n}\) is a unit vector perpendicular to both \( \textbf{A} \) and \( \textbf{B}.\) In component form, if \( \textbf{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \) and \( \textbf{B} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k},\) their cross product is: \[ \textbf{A} \times \textbf{B} = (a_2 b_3 - a_3 b_2) \hat{i} - (a_1 b_3 - a_3 b_1) \hat{j} + (a_1 b_2 - a_2 b_1) \hat{k}\] If \( \textbf{A} \times \textbf{B} = 0,\) it implies \( \textbf{A} \) and \( \textbf{B} \) are parallel or one of the vectors is the zero vector.

Perpendicular Vectors

Perpendicular vectors, or orthogonal vectors, meet at a right angle (90 degrees). The key property of perpendicular vectors is that their dot product is zero. That means \( \textbf{A} \cdot \textbf{B} = 0,\) where \( \textbf{A} \) and \( \textbf{B} \) are the vectors in question. It is crucial to note that even though the dot product is zero, the individual vectors need not be zero. For instance, if \( \textbf{A} = 2\textbf{i} - 3\textbf{j} + \textbf{k}\) and \( \textbf{B} = \textbf{i} + 3\textbf{j} - 2\textbf{k},\) then \( \textbf{A} \cdot \textbf{B} = 0\) illustrating they are orthogonal, but neither vector is zero.

Parallel Vectors

Vectors are parallel if they have the same or exact opposite direction. This means one vector is a scalar multiple of the other. The cross product of parallel vectors is zero, that is, \( \textbf{A} \times \textbf{B} = 0.\) This can be seen because the sine of the angle between them is zero (since the angle is either 0 or 180 degrees). For example, if \( \textbf{A} = 2\textbf{i} - 3\textbf{j} + \textbf{k}\) and \( \textbf{B} = k \textbf{A} = k \times (2\textbf{i} - 3\textbf{j} + \textbf{k})\) for any scalar \( k,\) then \( \textbf{A} \times \textbf{B} = 0.\) It’s essential to understand that if both the dot product and cross product of two vectors are zero \( (\textbf{A} \cdot \textbf{B} = 0 \text{ and } \textbf{A} \times \textbf{B} = 0),\) then \( \textbf{B}\) must be the zero vector.