Question

Derive an expression for a scalar product of two vectors in terms of their scalar components. Give two examples of scalar products.

Answer 1

Hint: Assume any vectors and multiply them. If you observe the final result, the value we get will be a scalar. We can say it is a scalar if we verify if the value shows any direction or just the magnitude of the result.

Complete answer:
Let us assume two vectors $\vec{p},\vec{q}$and they are equal to $(a\hat{i}+b\hat{j});(c\hat{i}+d\hat{j})$respectively.
If we dot product the two vectors assumed above,
We get,
$\begin{align}
  & \vec{p}.\vec{q}=(a\hat{i}+b\hat{j}).(c\hat{i}+d\hat{j}) \\
 & \vec{p}.\vec{q}=(a.c)+(b.d)+0 \\
 & \vec{p}.\vec{q}=ac+bd \\
\end{align}$
Clearly, we can see, the result that we got after doing dot product to both the vectors is a scalar. The result doesn’t depict any direction, it only represents the magnitude of the product.
Therefore, we can conclude that the product of two vectors is always a scalar.

Additional Information:
A scalar quantity is defined as the physical quantity that only has magnitude but doesn’t have any direction. Best example for a scale quantity is mass or electric charge. Coming to vector, vector is defined as the physical quantity that has both magnitude as well as direction. Best examples for vectors are force, weight, electric field. Scalar and vector products are two ways to multiply two different vectors. They are mostly used in physics and astronomy. The scalar product of two vectors is defined as the product of magnitudes of two vectors and the cosine of the angle between them. The scalar product of a vector and itself is a positive real number.

Note:
The scalar product is sometimes misunderstood as the inner product of two vectors. Inner product of two vectors can be multiplication of a scalar with a vector which results in a vector. But the scalar product is always a scalar. The scalar product of two vectors gives a scalar whereas the vector product of two vectors will always give a vector.