Scalar product

• Scalar product, also known as dot product, is a binary operation that takes two vectors and returns a scalar.

• This operation is defined as the product of the lengths (magnitudes) of the two vectors and the cosine of the angle between them. Symbolically, this can be written as: a·b =

  a

  b

  cos θ

• For two vectors a and b in standard position, you can also compute the scalar product as the sum of the products of their corresponding components. If a = (a1, a2) and b = (b1, b2), then a·b = a1b1 + a2b2.
• Scalar product has the property of commutativity, which means the result is the same irrespective of the order of multiplication. So, a·b = b·a.
• It also obeys distributivity over vector addition. This means that (a + b)·c = a·c + b·c.

• The scalar product of a vector with itself gives the square of its magnitude. That is, a·a =

  a

  ^2.

• The scalar product of two perpendicular vectors is zero because the cosine of 90 degrees is zero.
• Scalar product is widely used in physics to find the work done by a force or to project a vector onto another vector. Calculating the scalar product can help determine whether two given vectors are orthogonal, or at 90 degrees to one another.
• When calculating the scalar product, be sure to check the angle between the vectors. If vectors are given in component form, use the component method for the calculation.
• The units of the scalar product will be the product of the units of the two vectors. For example, if the vectors represent force (Newton) and displacement (metres), the scalar product will have units of energy (Joules).
• To revise scalar product, practice examples of finding dot products both graphically and algebraically, as well as solving problems involving cosine of the angle between two vectors. Also, understand its applications in various physical phenomena.