Question
Question: If an unit vector \(a\hat{i}+b\hat{j}\)is perpendicular to \(\left( \hat{i}-\hat{j} \right)\) then t...
If an unit vector ai^+bj^is perpendicular to (i^−j^) then the value of a and b are : A).$\dfrac{1}{\sqrt{2}},\dfrac{1}{\sqrt{2}}$$$$$ B). $\dfrac{1}{\sqrt{2}},-\dfrac{1}{\sqrt{2}}$$$$$ C). $-\dfrac{1}{\sqrt{2}},-\dfrac{1}{\sqrt{2}}$$$$$ D). Cannot be determined
Solution
We use the condition of two vectors a=a1i^+a2j^,b=b1i^+b2j^ perpendicular that is a1b1+a2b2=0 to find a relation between to express b in terms of a. We use the magnitude of a unit vector as 1 to make the equation , put b in terms of a and then solve for a. We then find b. $$$$
Complete step-by-step solution:
We know that the unit vector is a vector with magnitude 1. We also know that i^ and j^ are unit vectors (vectors with magnitude 1) along x,y axes respectively. So the magnitudes of these vectors are i^=j^=1.
We know that any vector a on a plane can be expressed in terms of unit vectors with its components a1 along x−axis and a2 along y−axis respectively as
a=a1i^+a2j^
The magnitude of the vector a is given as
a=a12+a22
We know that if two vectors a=a1i^+a2j^,b=b1i^+b2j^ are perpendicular to each other then their sum of products of corresponding components is zero which means ;
a1b1+a2b2=0
We are given in the question that the unit vector ai^+bj^is perpendicular to (i^−j^). Lee us denote the unit vector as n^=ai^+bj^ and the vector as v=i^−j^. The x−components of n^ and v are a and 1 respectively. The y−components of f n^ and v are b and 1 respectively. We use the perpendicular condition of two vectors and have;