Definition : Let \(\vec a \ and \ \vec b \) are two vectors whose magnitude are a and b respectively. If θ is the angle between \(\vec a \ and \ \vec b \), then ab cos θ is the scalar product of \(\vec a \ and \ \vec b \) and it is denoted by \(\vec a . \vec b. \)
So, scalar product of two vectors \(\vec a \ and \ \vec b \) is
\(\vec a . \vec b \)= |a ||b|cos θ, (0 < θ < π/2)
= ab cos θ
where \(\vec a = a \ and\ |\vec b| = b\)
scalar product of two vectors is equal to the product of their magnitude and cosine between the vectors. It is a real scalar number.
\(\vec a . \vec b \) is read as \(\vec a . \vec b. \)
If \(|\vec a|\) = 0 or \(|\vec b|\) = 0, then θ is not defined because zero product has no direction.
In this case
\(\vec a . \vec b = 0\)
Let \(\vec a \ and \ \vec b \) are two non-zero vectors then \(\vec a . \vec b. \) = 0 if and only if a and b are mutually perpendicular.
We know that unit vectors along axes are i,j and k.
Now î .î = |î| |î| cos0° = (1) (1) (1)= 1
Similarly ĵ.ĵ = 1 and k̂.k̂ = 1
Thus î. î = ĵ.ĵ = k̂. k̂ = 1
Now î .ĵ = |î||ĵ| cos π/2
(Angle between X and Y axes is π/2)
= (1) (1) (0) = 0
Similarly ĵ.k̂ = 0 and k̂.î = 0
î.ĵ = ĵ.k̂ = k̂.î = 0
If angle between two non-zero vector \(\vec a \ and \ \vec b \) is θ, then
i.e., scalar product of two vectors is commutative.
