Angle between Vectors MCQ Quiz in বাংলা - Objective Question with Answer for Angle between Vectors - বিনামূল্যে ডাউনলোড করুন [PDF]

Concept:

Let

The magnitude of the vector of a =

Unit vector =

Let the two vectors be  then dot Product of two vectors:

Where, || = Magnitude of vectors a and || = Magnitude of vectors b and θ is the angle between a and b

Calculation:

Given:  are unit vectors

Now,

 = 1 + 1 - 2 × 1 × 1 cos θ

= 2 - 2 cos θ

= 2 (1 - cos θ)

= 2 × 2 sin² (θ/2)

∴

⇒

Solution:

Given that, \(|\overrightarrow{a}| = 1, |\overrightarrow{b}| = 3\) and \(|\overrightarrow{a} - \overrightarrow{b}| = \sqrt{7}\)

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Hence, The angle between  and   is 

∴ The correct option is (4)

Concept:

Angle between vectors  is θ

then 

If  are unit vectors, then we have 

Calculation:

Given: that  are two unit vectors

We know that 

Taking square root on both sides.

Concept:

• The angle between two vectors   and  is given by, ​ 
• If  = a1 + a2 + a3 and   = b1 + b2 + b3, then  
• If position vectors of A and B are a1 + a2 + a3 and b1 + b2 + b3 respectively, then the vector AB = (b1 - a1) + (b2 - a2) + (b3 - a3)

Calculation:

Given: The position vectors of vertices A, B and C of triangle ABC are respectively  and .

⇒  = 

  = 

  = 

The angle between the vectors  and  is angle C

⇒ 

⇒  = 0

⇒ cos C = 0

⇒ C = 

∴ The correct option is (4).

Concept:

1) Let the two vectors be  and  then the angle between  and  is given by 

2) Magnitude of vector  aî + bĵ + ck̂ is 

Calculation:

Let  and  

The angle between  and  is given by  :

⇒ θ =  = 

⇒ θ =  = 

The angle between the vectors î − ĵ and ĵ − k̂ is 2π/3.

The correct answer is option 2.

Concept:

Calculation:

Given  and  are unit vectors.

Therefore,  and 

⇒   ⋅  = cos θ       ...(1)

Given  is a unit vector.

⇒  = 1

⇒ = 1

⇒ 3 + 1 -  cos θ = 1  [From equation 1]

⇒ cos θ = 

⇒ cos θ = 

⇒ θ = 30^o

∴ The angle between  and  for  to be a unit vector is 30°.

The correct answer is option 1.

Concept:

If two vector  and  then 

Where θ is the angle between vector   and .

Consider vector   then magnitude of  =  |A| = 

Calculation:

Given that,

Taking Magnitude both sides, we get

⇒ 

⇒ 

⇒ 13 × 2 × sin θ = 

⇒ 13 × 2 × sin θ = 13

⇒ sin θ = 

∴ θ = 30°

Concept:

•  are two vectors perpendicular to each other ⇔ 

Calculation:

Given  1) and  perpendicular

∴ 

⇒ 6 + 6 – 4λ = 0

⇒ 12 – 4λ = 0

∴ λ = 3

Concept Used:

If the angle between two vector  and  is θ.

then  and 

where || represents the magnitude of the vector.

Calculation:

Let three unit vectors be ,  and  such that 

⇒ 

⇒ 

⇒ 

⇒  ⇒ θ = 

⇒ θ = 

​Hence, the required value is:​

​

⇒ 

⇒ 

⇒ 

Concept:

For calculating the angle between two vectors, we generally use the dot product method. The dot product of two vectors a and b is defined as:

where θ is the angle between vectors a and b.

Scalar Triple Product:

If any two of the are same then [a b c] = 0

Calculation:

We know that,

 is perpendicular to both  and .

⇒ (A + B) (A × B) cos α = (A + B).(A × B)

⇒ A.(A × B) + B.(A × B)

⇒ B.(A × A) + A.(B × B)

⇒ [B A  A] + [A  B B]

⇒ 0 + 0 =0

∴ cos α = 0

⇒ α = 90° 

Hence, The angle between vectors  and  is 90°.