Question

What is a dot product geometrically?

Answer 1

Hint: There are two products, dot product and cross product, where the dot product of two vectors gives a scalar quantity and the cross product of two vectors gives a vector quantity. The term geometrically refers that how two vectors are placed in geometrical representation as we know that even if the result of some equation gives some significance arithmetically, it can have a different significance geometrically or in the real world.

Formula Used: \[A \cdot B = |A||B|\cos \theta \], where A and B are any two vectors, same or different, and theta \[\left( \theta \right)\] is the angle made by those vectors.

Complete step-by-step solution:
The dot product of two vectors is equivalent to the product of the magnitude of two vectors and the cosine of the angle between them, as per the geometrical definition. If there are two vectors A and B and the angle between the vectors A and B is theta \[\left( \theta \right)\], then we can say that \[A \cdot B = |A||B|\cos \theta \].
We will make different cases for different values of theta \[\left( \theta \right)\] to see how it impacts the dot product of two vectors.

Case 1: Let the measurement of angle be \[\theta = {90^ \circ }\], where the vectors are orthogonal to each other.
We will get the dot product of two vectors A and B as,
\[
  A \cdot B = |A||B|\cos \theta \\
   \Rightarrow A \cdot B = |A||B|\cos {90^ \circ } \\
   \Rightarrow A \cdot B = |A||B|\left( 0 \right) \\
   \Rightarrow A \cdot B = 0
 \]

Case 2: Let the measurement of angle be \[\theta = {0^ \circ }\], where the vectors are parallel to each other.
We will get the dot product of two vectors A and B as,
\[
  A \cdot B = |A||B|\cos \theta \\
   \Rightarrow A \cdot B = |A||B|\cos {0^ \circ } \\
   \Rightarrow A \cdot B = |A||B|\left( 1 \right) \\
   \Rightarrow A \cdot B = |A||B|
 \]

So, geometrically the dot product of two vectors is equivalent to the product of the magnitude of two vectors and the cosine of the angle between them.

Note: The magnitude of any vector A is the containing vectors i, j, and k is found by taking the square of scalar quantities with vectors i, j, and k, and then taking its sum and then finally taking its root. If \[A = x\hat i + y\hat j + z\hat k\], then \[|A| = \sqrt {{x^2} + {y^2} + {z^2}} \].