Question

If the magnitude of two vectors are 4 and 6 and the magnitude of their scalar product is $ 12\sqrt 2 $ , then what is the angle between these vectors?

Answer 1

Hint: The dot product of two vectors is a scalar quantity, also referred to as the scalar product, as in this case. The value of the dot product depends on the cosine of the angle between them.

Formula Used: The formulae used in the solution are given here.
 $ a \cdot b = \left| a \right|\left| b \right|\cos \theta $ where $ \theta $ is the angle between the two vectors $ a $ and $ b $ .

Complete step by step answer:
A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its head.
The dot product of two vectors always results in scalar quantity, i.e. it has only magnitude and no direction. It is represented by a dot (.) in between two vectors.
a dot b = $ a \cdot b $
The mathematical value of the dot product is given as
 $ a \cdot b = \left| a \right|\left| b \right|\cos \theta $ where $ \theta $ is the angle between the two vectors $ a $ and $ b $ .
It has been given that the magnitude of two vectors are 4 and 6 and the magnitude of their scalar product is $ 12\sqrt 2 $ .
Let the two vectors be $ m $ and $ n $ . It has been given in the question that, $ \left| m \right| = 4 $ and $ \left| n \right| = 6 $ .
 The scalar product between these vectors is given by,
 $ m \cdot n = \left| m \right|\left| n \right|\cos \theta $ where $ \theta $ is the angle between the two vectors $ m $ and $ n $ .
Given, $ m \cdot n = 12\sqrt 2 $ . Thus we can write.
 $ 12\sqrt 2 = 6 \cdot 4\cos \theta $
 $ \Rightarrow \cos \theta = \dfrac{{12\sqrt 2 }}{{24}} = \dfrac{1}{{\sqrt 2 }} $
Now we know that, $ \cos {45^ \circ } = \dfrac{1}{{\sqrt 2 }} $ .
Therefore, the angle between the two vectors is $ \theta = {45^ \circ } $ .

Note:
The cross product of two vectors results in a vector quantity. It is represented by a cross sign between two vectors.
 $ a \times b $
The mathematical value of a cross product-
 $ a \times b = \left| a \right|\left| b \right|\sin \theta \hat n $
where, $ \left| a \right| $ is the magnitude of vector $ a $ , $ \left| b \right| $ is the magnitude of vectors $ b $ , $ \theta $ is the angle between the two vectors $ a $ and $ b $ and $ \hat n $ is a unit vector showing the direction of the multiplication of two vectors.