Home
Class 12
PHYSICS
Two vectors vec(A) and vec(B) lie in X-Y...

Two vectors `vec(A)` and `vec(B)` lie in X-Y plane. The vector `vec(B)` is perpendicular to vector `vec(A)`. If `vec(A) = hat(i) + hat(j)`, then `vec(B)` may be :

A

`hat(i) - hat(j)`

B

`-hat(i) + hat(j)`

C

`-2hat(i) + 2 hat(j)`

D

Any of the above

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to find the vector \(\vec{B}\) that is perpendicular to the vector \(\vec{A} = \hat{i} + \hat{j}\). ### Step-by-Step Solution: 1. **Understanding Perpendicular Vectors**: Two vectors \(\vec{A}\) and \(\vec{B}\) are perpendicular if their dot product is zero. Mathematically, this is expressed as: \[ \vec{A} \cdot \vec{B} = 0 \] 2. **Calculating the Dot Product**: Given \(\vec{A} = \hat{i} + \hat{j}\), we can express \(\vec{B}\) in general form as \(\vec{B} = a\hat{i} + b\hat{j}\), where \(a\) and \(b\) are the components of vector \(\vec{B}\). The dot product \(\vec{A} \cdot \vec{B}\) is calculated as: \[ \vec{A} \cdot \vec{B} = (\hat{i} + \hat{j}) \cdot (a\hat{i} + b\hat{j}) = \hat{i} \cdot a\hat{i} + \hat{j} \cdot b\hat{j} \] \[ = a + b \] 3. **Setting the Dot Product to Zero**: For \(\vec{A}\) and \(\vec{B}\) to be perpendicular, we set the dot product to zero: \[ a + b = 0 \] 4. **Finding Possible Values for \(\vec{B}\)**: From the equation \(a + b = 0\), we can express \(b\) in terms of \(a\): \[ b = -a \] Thus, \(\vec{B}\) can be written as: \[ \vec{B} = a\hat{i} - a\hat{j} = a(\hat{i} - \hat{j}) \] This means that any vector of the form \(k(\hat{i} - \hat{j})\) (where \(k\) is any scalar) will be perpendicular to \(\vec{A}\). 5. **Checking Options**: We can check the given options to see which ones satisfy the condition \(a + b = 0\): - **Option 1**: \(\vec{B} = \hat{i} - \hat{j}\) → \(1 - 1 = 0\) (Correct) - **Option 2**: \(\vec{B} = -\hat{i} + \hat{j}\) → \(-1 + 1 = 0\) (Correct) - **Option 3**: \(\vec{B} = -2\hat{i} + 2\hat{j}\) → \(-2 + 2 = 0\) (Correct) - **Option 4**: Any of the above (Correct) ### Final Answer: All options provided are correct since they all yield a dot product of zero with \(\vec{A}\).
Promotional Banner

Similar Questions

Explore conceptually related problems

Let a vector vec(a) be coplanar with vectors vec(b) = 2hat(i) + hat(j) + hat(k) and vec(c) = hat(i) - hat(j) + hat(k) . If vec(a) is perpendicular to vec(d) = 3hat(i) + 2hat(j) + 6hat(k) , and |vec(a)| = sqrt(10) . Then a possible value of [[vec(a),vec(b),vec(c)]] + [[vec(a), vec(b), vec(d)]] + [[vec(a),vec(c),vec(d)]] is equal to :

If vec(A) and vec(B) are perpendicular Vectors and vector vec(A)= 5hat(i)+7hat(j)-3hat(k) and vec(B)= 2hat(i)+2hat(j)-ahat(k) . The value of a is

Given: vec(A)=Acos theta hat(i)+Asin theta hat(j) . A vector vec(B) , which is perpendicular to vec(A) ,is given by

If for two vectors hat(A) and hat(B) ,sum (vec(A)+vec(B)) is perpendicular to the diffrence (vec(A)-vec(B)) . Find the ratio of their magnitude.

There are two vector vec(A)=3hat(i)+hat(j) and vec(B)=hat(j)+2hat(k) . For these two vectors- (i) Find the component of vec(A) along vec(B) and perpendicular to vec(B) in vector form. (ii) If vec(A) & vec(B) are the adjacent sides of parallelogram then find the magnitude of its area. (iii) Find a unit vector which is perpendicular to both vec(A) & vec(B) .

If vec a is a unit vector such that vec a xxhat i=hat j then find vec a

Given vec(A) = 3hat(i) + 2hat(j) and vec(B) = hat(i) + hat(j). The component of vector vec(A) along vector vec(B) is

Given two vectors vec(A)=3hat(i)+hat(j)+hat(k) and vec(B)=hat(i)-hat(j)-hat(k) .Find the a.Area of the triangle whose two adjacent sides are represented by the vector vec(A) and vec(B) b.Area of the parallelogram whose two adjacent sides are represented by the vector vec(A) and vec(B) c. Area of the parallelogram whose diagnoals are represented by the vector vec(A) and vec(B)

Let vec u be a vector coplanar with the vectors vec a=2hat i+3hat j-hat k and vec b=hat j+hat k If vec u is perpendicular to vec a and vec u.vec b=24 then |vec u|^(2) is equal to

Let vec(a) = hat(i) + hat(j) + hat(k), vec(b) and vec(c )= hat(j)-hat(k) be three vectors such that vec(a) xx vec(b)= vec(c ) and vec(a).vec(b)=1 . If the length of projection vector of the vector vec(b) on the vector vec(a) xx vec(c ) is l, then the vlaue of 3l^(2) is equal to ____