Question

Two vectors, in three dimensions, are given in Cartesian coordinates as :

Answer 1

The angle between the two vectors is 0.365 rad

Explanation:

The angle between two 3-D vectors can be found by using the following relationship:

$a\cdot b = |a||b|cos \theta$

where

$a\cdot b$ is the scalar product between the two vectors

$|a|$ is the magnitude of vector a

$|b|$ is the magnitude of vector b

$\theta$ is the angle between the two vectors

The two vectors in this problem are

a = (8,8,9)

b = (4,9,5)

So their scalar product is

$a\cdot b = 8\cdot 4 + 8\cdot 9 + 9\cdot 5=32+72+45=149$

The magnitude of a is

$|a|=\sqrt{8^2+8^2+9^2}=\sqrt{64+64+81}=14.5$

The magnitude of b is

$|b|=\sqrt{4^2+9^2+5^2}=\sqrt{16+81+25}=11.0$

So now we can re-arrange the equation to find the angle between the two vectors:

$cos \theta = \frac{a\cdot b}{|a||b|}=\frac{149}{(14.5)(11.0)}=0.934\\\theta=cos^{-1}(0.934)=0.365 rad$

Learn more about operations with vectors:

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