Unit Vectors MCQ Quiz - Objective Question with Answer for Unit Vectors - Download Free PDF

Explanation:

  and  are unit vectors

So, || = || = 1

Now, 

|| = 1 if

i.e., 

i.e., 

i.e., θ = 

Hence  is unit vector, if angle between  and  is 

Option (4) is true.

Explanation:

 and  are unit vectors.

So,  = 1

Also,  is perpendicular to both  and .

So,  is parallel to 

i.e.,  where λ is any scaler.

Given, angle between  and  is 30°.

So, 

⇒ 

⇒  (Taking mod both sides)

⇒ 

⇒ |λ| = 2

⇒ λ = ± 2

Hence we get

Option (3) is true.

Explanation:

  and  are unit vectors

So, || = || = 1

Now, 

|| = 1 if

i.e., 

i.e., 

i.e., θ = 

Hence  is unit vector, if angle between  and  is 

Option (4) is true.

Calculation:

 is a non-zero vector of modulus a and λ is a non-zero scalar and  is a unit vector i.e.  = 1

⇒ |λ| = 1   (∵ |mn| = |m|⋅|n|)

⇒ |λ|⋅a = 1   (∵  = a(given))

⇒ a = 

The correct answer is option "3"

Concept:

The unit vector of the vector  is 

Explanation:

 =  = 

|| = || = 

So, unit vector is  = 

Option (5) is true.

Concept:

Unit vector in the direction of vector is given by .

Calculation:

Given: 

⇒ Unit vector in the direction of vector  is given by .

⇒  

A vector in the direction of vector  that has magnitude 10 is given by 

⇒ 

Hence, option 2 is correct.

Concept:

Let  and  be the two vectors, then the vector  perpendicular to  both  

Calculation:

Here,  and 

Vector perpendicular to both vectors will be 

Hence, option (1) is correct.

Explanation:

Given, λî + 2λĵ + 2λk̂ ​is a unit vector.

∴ 

⇒ 

⇒ 

⇒ 

⇒ 3λ = 1

⇒ 

Concept:

• If a and b are any two unit vectors then  .

Calculations:

Given  and 

Consider,

  .....(1)

Substitute   and   in (1)

∴ 

Hence, the correct answer is option 2).

Concept:

The unit vector in the direction of vector  is given by .

Calculation:

Given:  & 

⇒  = 

Unit vector in the direction of vector  

⇒ 

⇒  

⇒  

Hence, option 4 is correct.

Concept:

1. The scalar product of two vectors a and b of magnitude |a| and |b| is given as |a||b| cos θ, where θ represents the angle between the vectors a and b taken in the direction of the vectors.
2. We can express the scalar product as =|a||b| cosθ, where |a| and |b| represent the magnitude of the vectors a and b while cos θ denotes the cosine of the angle between both the vectors and a.b indicates the dot product of the two vectors.

Calculation:

Given:

 and  be two unit vectors such that  and 2θ is the angle between  and 

⇒ 

We are given 

Squaring both sides, we get, 

⇒ 

⇒ 

⇒ 

⇒ 1 + 1 - 2 

⇒ 1 + 1 - 2​ . 1 . 1 . cos2θ

⇒ 2 - 2.cos2θ

⇒ 1 - cos2θ

⇒ 2 sin²θ 

⇒ sin2θ 

∴ -1≤ sin θ

Concept:

Vector  of magnitude |p| in the direction of  is given by

​If   then magnitude of ​ is written as​​

Calculation:

Let the required vector is 

A vector in the direction a and magnitude |b| is given by

Hence, option 4 is correct.

Concept:

If  perpendicular to both the vectors  and   then 

Unit vector,   

Calculation :

Here the given vectors are,   and  , 

Let, vector  are perpendicular to both the vectors, 

So,   

⇒  

⇒  

∴  =  

Hence the unit vector perpendicular to both the vectors are, 

 =  

⇒  

The correct option is 1. 

Calculation:

Given,  and    all unit vectors        ---(A)

And

    ---(1)

Take,

1 + 1 + 2 ab = 8 (ab  is unit vector)

2ab = 6

ab = 3

take,

b c = 3

Similarly, c a = 3

Take,

On squaring both sides we get,

i.e |2â + b̂ + ĉ| = 6

Concept:

• The unit vector û in the direction of a vector  is given by: , where  is the magnitude of the vector .
• The magnitude of a vector  is given as: 

.

Calculation:

We have  and .

∴ 

⇒ 

The magnitude of  will be:

Now, the unit vector along the direction of  will be:

⇒ .