Explain scalar product of two vectors. Mention its any two properties. For what value of 'a' `vecA = 2hati + a hatj + hatk` is perpendicular to `vecB = 4hati - 2hatj - 2hatk` ?

Explanation

Scalar product or Dot product : Scalar product of two vectors `vecA and vecB` is equal to product of magnitude of `vecA and vecB` multiplied by the cosine of the smaller angle between them. i.e., `vecA .vecB = |vecA||vecB| cos theta` , It gives projection of one vector in the direction of other.
Properties of dot product
i) Dot product is distributive over vector addition i.e., `vecA. .(vecB +vecC) = vecA. vecB + vecA . vecC`
ii) Dot product is commutative i.e., `vecA.vecB = vecB-vecA`
iii) Dot product of two vectors in components form is given by
`vecA. vecB = A_x B_x + A_y B_y + A_zB_z`
To find the value of a
Two vectors `vecA and vecB` will be perpendicular to each other if `vecA .vecB = 0 (2hati + ahatj + hatk) . (4hati - 2hatj - 2hatk) = 0`
`implies 8 - 2a - 2 = 0 implies 2a = 6 implies a = 3`
Thus, for `a = 3`, vector `vecA` is perpendicular to `vecB`.