Let's denote the unit vector as
=a+b, where
a and
b are the components of the vector along the
x and
y axes, respectively. According to the given problem, this unit vector makes an angle of
45∘ with
+ and an angle of
60∘ with
3−4.
We can start by using the dot product formula to generate necessary equations. The dot product of two vectors
and
is given by:
⋅=||||cosθSince
is a unit vector,
||=1, and thus the dot product simplifies to:
⋅=||cosθLet's apply this to both given vectors.
For the first vector,
+ :
Here,
1=+ and the angle is
45∘.
The magnitude of
1 is:
|1|=√12+12=√2The equation becomes:
⋅(+)=√2cos45∘a+b=√2⋅a+b=1 For the second vector,
3−4 :
Here,
2=3−4 and the angle is
60∘.
The magnitude of
2 is:
|2|=√32+(−4)2=√9+16=√25=5The equation becomes:
⋅(3−4)=5cos60∘3a−4b=5⋅3a−4b= Now we have a system of two equations:
a+b=13a−4b=We can solve this system of equations to find the values of
a and
b.
From the first equation, solve for
b :
b=1−aSubstitute this into the second equation:
3a−4(1−a)=3a−4+4a=7a−4=7a=+47a=+7a=a= Substitute back to find
b :
b=1−b=−b=Thus, the unit vector is:
=+Therefore, the correct option is:
Option A:
+But
||=√()2+()2≠1. So,
is not an unit vector.