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Evaluate the product $(3\overrightarrow{a}-5\overrightarrow{b}).(2\overrightarrow{a}+7\overrightarrow{b})$ .

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Hint: We have to evaluate $(3\overrightarrow{a}-5\overrightarrow{b}).(2\overrightarrow{a}+7\overrightarrow{b})$ , so multiply each other in step by step manner. After that, while simplifying use the properties $\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{a}$ and $\overrightarrow{a}.\overrightarrow{a}={{\left| \overrightarrow{a} \right|}^{2}}$ . Try it, you will get the answer.

Complete step-by-step answer:
Vector is an object which has magnitude and direction both. It is represented by a line with an arrow, where the length of the line is the magnitude and the arrow shows the direction. We can consider any two vectors as equal if their magnitude and direction are the same. It plays an important role in Mathematics, Physics as well as in Engineering. It is also known as Euclidean vector or Geometric vector or Spatial vector or simply “vector“. According to vector algebra, a vector can be added to the other vector. Let us have a detailed discussion of vector math with its definition, representation, magnitude and its operations.
The vectors are defined as an object containing both magnitude and direction. Vector describes the movement of an object from one point to another. Vector math can be geometrically picturised by the directed line segment. The length of the segment of the directed line is called the magnitude of a vector and the angle at which the vector is inclined shows the direction of the vector. The beginning point of a vector is called “Tail” and the end side (having an arrow) is called “Head.”
A vector math is defined as mathematical structure. It has many applications in physics and geometry. We know that the location of the points on the coordinate plane can be represented using the ordered pair such as $(x,y)$ . The usage of vectors are very useful in the simplification process of three-dimensional geometry. Along with the term vector, we have heard the term scalar. A scalar actually represents the “real numbers”. In simpler words, a vector of “ $n$ ” dimensions is an ordered collection of n elements called “components“.
$\begin{align}
  & (3\overrightarrow{a}-5\overrightarrow{b}).(2\overrightarrow{a}+7\overrightarrow{b})=3\overrightarrow{a}.2\overrightarrow{a}+3\overrightarrow{a}.7\overrightarrow{b}-5\overrightarrow{b}.2\overrightarrow{a}-5\overrightarrow{b}.7\overrightarrow{b} \\
 & =6(\overrightarrow{a}.\overrightarrow{a})+21(\overrightarrow{a}.\overrightarrow{b})-10(\overrightarrow{b}.\overrightarrow{a})-35(\overrightarrow{b}.\overrightarrow{b}) \\
 & =6(\overrightarrow{a}.\overrightarrow{a})+21(\overrightarrow{a}.\overrightarrow{b})-10(\overrightarrow{a}.\overrightarrow{b})-35(\overrightarrow{b}.\overrightarrow{b}) \\
 & =6(\overrightarrow{a}.\overrightarrow{a})+11(\overrightarrow{a}.\overrightarrow{b})-35(\overrightarrow{b}.\overrightarrow{b}) \\
 & =6{{\left| \overrightarrow{a} \right|}^{2}}+11(\overrightarrow{a}.\overrightarrow{b})-35{{\left| \overrightarrow{b} \right|}^{2}} \\
\end{align}$ …(Using property $\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{a}$ and $\overrightarrow{a}.\overrightarrow{a}={{\left| \overrightarrow{a} \right|}^{2}}$ )
Therefore, $(3\overrightarrow{a}-5\overrightarrow{b}).(2\overrightarrow{a}+7\overrightarrow{b})==6{{\left| \overrightarrow{a} \right|}^{2}}+11(\overrightarrow{a}.\overrightarrow{b})-35{{\left| \overrightarrow{b} \right|}^{2}}$ .
Note: Read the question carefully. Also, take utmost care that no terms are missing. Do not make silly mistakes while solving. While simplifying, take care that you solve it step by step. Do not confuse while solving.

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