Consider the following statements:

1. The cross product of two unit vectors is always a unit vector.

2. The dot product of two unit vectors is always unity.

3. The magnitude of sum of two unit vectors is always greater than the magnitude of their difference.

Which of the above statements are not correct?

Concept:

The cross product of two vectors \(\vec a \ and \ \vec b \)is given by \(\vec a \times \vec b = \left| {\vec a} \right| \cdot \left| {\vec b} \right|\sin θ \;\hat n\) and \(|\vec a \times \vec b| = \left| {\vec a} \right| \cdot \left| {\vec b} \right|\sin θ \;\)

The scalar product of two vectors \(\vec a \ and \ \vec b \)is given by \(\vec a \cdot \;\vec b = \left| {\vec a} \right| \times \left| {\vec b} \right|\cos θ \)

If \(\vec a\) is a unit vector then \(|\vec a| = 1\)

Calculations:

Statement 1: The cross product of two unit vectors is always a unit vector.

Let \(\vec a\) and \(\vec b\) are two unit vectors.

i.e \(|\vec a| = 1 \ and \ |\vec b| = 1\)

As we know that, the cross product of two vectors \(\vec a \ and \ \vec b \)is given by \(\vec a \times \vec b = \left| {\vec a} \right| \cdot \left| {\vec b} \right|\sin θ \;\hat n\) and \(|\vec a \times \vec b| = \left| {\vec a} \right| \cdot \left| {\vec b} \right|\sin θ \;\)

⇒ \(|\vec a \times \vec b| = \left| {\vec a} \right| \cdot \left| {\vec b} \right|\sin θ \; = sin θ \)

The range of sin θ is [-1, 1]

So, it is not necessarily true that the cross product of two unit vectors is always a unit vector.

Hence, statement 1 is false.

Statement 2: The dot product of two unit vectors is always unity.

Let \(\vec a\) and \(\vec b\) are two unit vectors.

i.e \(|\vec a| = 1 \ and \ |\vec b| = 1\)

As we know that, the scalar product of two vectors \(\vec a \ and \ \vec b \)is given by \(\vec a \cdot \;\vec b = \left| {\vec a} \right| \times \left| {\vec b} \right|\cos θ \) 

⇒ \(|\vec a \cdot \;\vec b |= cos \ θ \)

The range of cos θ is [-1, 1].

So, it is not necessarily true that the dot product of two unit vectors is always a unit vector.

Hence, statement 2 is false.

Statement 3: The magnitude of sum of two unit vectors is always greater than the magnitude of their difference.

Let \(\vec a \ = \hat i \ and \ \ \vec b = \hat j\) 

As we can see that, the vectors  \(\vec a\) and \(\vec b\) are two unit vectors

⇒  \(|\hat i + \hat j| = \sqrt 2\) and \(|\hat i - \hat j| = \sqrt 2\)

⇒ \(|\vec a + \vec b| = |\vec a - \vec b|\)

So, statement 3 is also false.

Hence, the correct option is 4.