Concept: The cross product of two vectors
and
is given by
×=|a|⋅|b|sin‌θ‌‌‌ and
|×|=|a|⋅|b|sin‌θ The scalar product of two vectors
and
is given by
⋅=|a|×|b|cos‌θ If
is a unit vector then
|a|=1 Calculations: Statement 1: The cross product of two unit vectors is always a unit vector.
Let
and
are two unit vectors.
i.e
∣a]=1 and
||=1 As we knowthat, the cross product of two vectors
and
is given by
×=∣a]⋅|b|sin‌θ‌‌‌ and
|×|=|a| ⇒|×|=|a|⋅|b|sin‌θ=sin‌θ The range of
sin‌θ is
[−1,1] So, it is notnecessarily 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
and
are two unit vectors.
i.e
|a|=1 and
∣b∣=1 As we know that, the scalar product of two vectors
and
is given by
⋅=||×|b|cos‌θ ⇒|⋅|=cos‌θ The range of
cos‌θ 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
= and
= As we can see that, the vectors
and
are two unit vectors
⇒|+|=√2‌ and ‌|−|=√2 ⇒|+|=|−| So, statement 3 is also false.
Hence, the correct option is 4 .