The result of scalar product and the vector product of two given vectors is zero . If one vector is `hat(i)` what is the other vector ?

Answer

To solve the problem step by step, we need to analyze the conditions under which both the scalar product (dot product) and the vector product (cross product) of two vectors yield zero.

### Step 1: Understand the Given Information
We are given one vector, which is \( \hat{i} \) (or \( \mathbf{A} = \hat{i} \)), and we need to find another vector \( \mathbf{B} \) such that both the scalar product and vector product of \( \mathbf{A} \) and \( \mathbf{B} \) are zero.

### Step 2: Scalar Product Condition
The scalar product (dot product) of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by:
\[ \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta \]
where \( \theta \) is the angle between the two vectors. For the scalar product to be zero, either one of the vectors must be the zero vector, or the angle \( \theta \) must be \( 90^\circ \) (i.e., the vectors are perpendicular).

### Step 3: Vector Product Condition
The vector product (cross product) of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by:
\[ \mathbf{A} \times \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \sin \theta \hat{n} \]
where \( \hat{n} \) is a unit vector perpendicular to the plane formed by \( \mathbf{A} \) and \( \mathbf{B} \). For the vector product to be zero, either one of the vectors must be the zero vector, or the vectors must be parallel (i.e., \( \theta = 0^\circ \) or \( 180^\circ \)).

### Step 4: Analyze the Given Vector
The given vector is \( \mathbf{A} = \hat{i} \), which can be expressed in component form as:
\[ \mathbf{A} = 1 \hat{i} + 0 \hat{j} + 0 \hat{k} \]

### Step 5: Determine the Other Vector
To satisfy both conditions (scalar and vector product being zero), the simplest solution is to let the other vector \( \mathbf{B} \) be the zero vector:
\[ \mathbf{B} = 0 \hat{i} + 0 \hat{j} + 0 \hat{k} = \mathbf{0} \]

### Step 6: Verify the Conditions
1. **Scalar Product**:
\[ \mathbf{A} \cdot \mathbf{B} = \hat{i} \cdot \mathbf{0} = 0 \]
2. **Vector Product**:
\[ \mathbf{A} \times \mathbf{B} = \hat{i} \times \mathbf{0} = \mathbf{0} \]

Both products yield zero, confirming that our solution is correct.

### Conclusion
The other vector \( \mathbf{B} \) is the zero vector:
\[ \mathbf{B} = \mathbf{0} \]