Exam Questions - Scalar product

Scalar Product

The scalar product of two vectors, also known as the dot product, amounts to the product of the magnitudes of the two vectors and the cosine of the angle between them.
When the vectors are represented in terms of components, the scalar product is the sum of the products of corresponding components.
If a and b are vectors, then **a · b = a b cosθ**, where θ is the angle between the vectors.

The scalar product is commutative, which means that a · b = b · a.
The scalar product is distributive over vector addition. That means a · (b + c) = a · b + a · c.
The scalar product of a vector with itself is the square of its magnitude. That is, **a · a = a ²**.
Two vectors are perpendicular if and only if their scalar product is zero.

In the cartesian form, a · b = a1b1 + a2b2 + a3b3, where a1, a2, a3, b1, b2, b3 are the components of the vectors.
In the component form, multiply corresponding components of the vectors and sum the results to get the scalar product.

Scalar product is used to compute the angle between two vectors. Knowing the scalar product and magnitudes of two vectors allows for the calculation of the cosine of the angle between them.
It is also used in determining whether two vectors are perpendicular. If their scalar product is zero, then the vectors are perpendicular.
In physics, scalar product finds use in calculating work done, which is the dot product of force and displacement vectors.

Given two vectors a = 3i + 4j and b = 5i - 2j, calculate a · b.
Find the angle between vectors a = 7i + 2j and b = -3i + 6j.
Determine whether vectors a = 2i + 3j + k and b = 5i - j + 3k are perpendicular.