Question

Is dot product a scalar?

Answer 1

Step 1: Explanation:

1. A number (a scalar quantity) is generated by performing a special operation on the vector components and is known as the dot product, also known as the scalar product of two vectors.
2. When using the dot product only pairs of vectors with the same dimensions are of any importance.
3. The scalar product of two vectors $a$ and $b$ with magnitude $\left|a\right|$ and $\left|b\right|$ is given as $\left|a\right|\left|b\right|\cos \theta$, where $\theta$ denotes the angle formed between the vectors $a$ and $b$ when taken in their respective directions.

Step 2: Example:

Example: If $a=\left(i+j+k\right)\&b=\left(2i+2j+k\right)$, then find $a.b$, where $Cos\theta =1$.

Solution:

$$\begin{gathered} a.b=\left(i+j+k\right).\left(2i+2j+k\right)Cos\theta \\ \Rightarrow a.b=\left(2+2+1\right)\times 1 \\ \Rightarrow a.b=5 \end{gathered}$$

Hence, the dot product is a scalar.