If vector a and b are two non collinear unit vectors and |vector a+ vector b|=√3 then find the value of ( vector a- vector b).(2 vector a+ vector b)
There is probably a more elegant way to solve the problem, but here's one that works.
We know a and b are unit vectors.
By the cosine rule, we can find the angle between the two vectors a+b as:
cos(φ)=(1^2+1^2-(√3)^2)/(2*1*1)=-1/2
=> φ=2π/3.
Without loss of generality (WLOG), we can assume a=<1,0>, from which we determine b=<1/2,(√3)/2>
from there, you can find the dot product as required.
To find the value of (vector a - vector b).(2 vector a + vector b), we can use the dot product formula:
(vector a - vector b).(2 vector a + vector b) = (vector a).(2 vector a + vector b) - (vector b).(2 vector a + vector b)
Let's calculate each term step by step.
Step 1: Calculate (vector a).(2 vector a + vector b)
To find the dot product, we multiply the corresponding components of the vectors and then add them together.
(vector a).(2 vector a + vector b) = 2(vector a).(vector a) + (vector a).(vector b)
Since vector a is a unit vector, its magnitude is 1. So, (vector a).(vector a) = |vector a|^2 = 1^2 = 1.
Also, since vector a and vector b are non-collinear, their dot product is zero: (vector a).(vector b) = 0.
Therefore, (vector a).(2 vector a + vector b) = 2(1) + 0 = 2.
Step 2: Calculate (vector b).(2 vector a + vector b)
Again, we multiply the corresponding components of the vectors and then add them together.
(vector b).(2 vector a + vector b) = 2(vector b).(vector a) + (vector b).(vector b)
Similarly to earlier, (vector b).(vector a) = 0 because vector a and vector b are non-collinear.
For the same reason, (vector b).(vector b) = |vector b|^2 = 1^2 = 1.
Therefore, (vector b).(2 vector a + vector b) = 2(0) + 1 = 1.
Step 3: Calculate the final result
Now we can substitute the values we obtained in steps 1 and 2 into the original expression.
(vector a - vector b).(2 vector a + vector b) = (vector a).(2 vector a + vector b) - (vector b).(2 vector a + vector b)
= 2 - 1
= 1.
So, the value of (vector a - vector b).(2 vector a + vector b) is 1.