Scalar or Dot Product MCQ Quiz - Objective Question with Answer for Scalar or Dot Product - Download Free PDF

Calculation:

Given,

Using the identity , we substitute:

Simplifying the equation:

Rearrange to isolate the sine terms:

Now, calculate the dot product:

∴ The value of is 2.

Hence, the correct answer is Option 4.

Calculation: 

⇒ 

⇒ 

⇒ 

⇒ 

Hence, the correct answer is Option 1.

Explanation:

Given:

 and  are perpendicular vectors.

⇒  = 0

⇒ 

⇒ 

Now  are unit vectors

⇒ 

⇒ 1.1 cosθ =1/2

⇒Cosθ = 1/2 

Now

cos2 θ = Cos²θ -1

= 2× 1/4 -1

= 

Now, 

cosθ + cos2θ = 1/2 -1/2 =0

∴The Correct answer is Option a

Concept:

​

Vector a is perpendicular to b if 

Calculation:

Squaring both sides,

Now we have,

∴  Vector a is perpendicular to b

Hence, option (3) is correct.

Calculation

Given: and  

⇒ 

⇒ .

Hence option 2 is correct.

Concept:

Dot Product: it is also called the inner product or scalar product

Let the two vectors are  and 

Dot Product of two vectors is given by:   = |a||b| cos θ

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

Formulas of Dot Product:

Calculation:

Given that,

(â + b̂ + ĉ) = 0    ----(1)

We know that,

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

⇒ (â + b̂ + ĉ)² = â ⋅ â + b̂ ⋅ b̂ + ĉ ⋅ ĉ + 2(â ⋅ b̂ + b̂ ⋅ ĉ + ĉ ⋅ â) 

From equation (1), we get

⇒ (1 + 1 + 1) + 2 (â ⋅ b̂ + b̂ ⋅ ĉ + ĉ ⋅ â) = 0

∴ (â ⋅ b̂ + b̂ ⋅ ĉ + ĉ ⋅ â) = - 3/2        

Concept:

If   and  are mutually perpendicular unit vectors 

,  

Calculations:

Consider, 

Given   and  are mutually perpendicular unit vectors.

So,  

And    and  are unit vectors

So, 

= 

= 15 - 12 

= 3

Concept:

Dot Product of two vectors  and  is defined as , where  is the magnitude of vector .

.

Calculation:

We are given that "sum of two vectors  and  is a vector ".

⇒ 

Taking dot product of both sides with themselves, the magnitudes will still be equal:

⇒ 

⇒ 

Since , we get:

⇒ 

⇒ 

⇒ 

Now, 

= 

= 4 + 4 - (-4)

= 12

⇒ .

Concept:

If  are two vectors then 

Note: If vectors  are perpendicular to each other then 

Calculation:

Given: 

Let θ be the angle between the vector 

⇒ 

We know that, 

⇒ 

⇒ 1 = 3 cos θ 

⇒  

⇒ 

Hence, option 3 is correct.

CONCEPT:

• If  is a unit vector then 

CALCULATION:

Given:  and  is a unit vector

⇒ 

As we know that, 

⇒ 

As we know that, if  is a unit vector then 

⇒ 

Hence, correct option is 2.

Concept:

• The cross product of vector to itself = 0
• The cross product of collinear vectors = 0
• The dot product of collinear vectors = Product of their Magnitudes
• For dot product 
• For cross product  
• The unit vector in the direction of a  
• A vector in direction of  = (Magnitude of ) ×

Calculation:

Given:

⇒ 

⇒ 

⇒ 

It means the  and  are collinear vectors.

∴  

⇒  

⇒ \(\boldsymbol{\rm\vec a + \vec b = 2\hat i + 3\hat j + 4\hat k}\)

⇒  

⇒  

⇒ 

Concept:

Dot Product: it is also called the inner product or scalar product

Let the two vectors are  and 

Dot Product of two vectors is given by:   = |a||b| cos θ

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

Formulas of Dot Product:

Calculation:

Let 

Similarly, 

Therefore 

= xî + yĵ + zk̂ 

Hence, required value of

= (xî + yĵ + zk̂ ).(xî + yĵ + zk̂ )

= x² + y² + z² = |a|²

CONCEPT:

• Projection of a vector  on other vector  is given by: 

CALCULATION:

Given:  and 

Here, we have to find the projection of a vector  on other vector  is given by: 

⇒ 

⇒ 

Hence, option 3 is correct.

CONCEPT:

The scalar product of two vectors is given by 

If the vectors  are perpendicular then 

CALCULATION:

Given:  are two vectors such that 

⇒ 

⇒ 

⇒ 

∵ 

⇒ 

⇒ 

⇒ 

So,  must be perpendicular to 

Hence, correct option is 3.

Concept:

If  is perpendicular to , then 

. = 0

Calculation:

 +  = 0

Given: 

So, we can write

 +   = 0

 = -       -----(1)

To find: 

 = 4 +       ----(2)

From equations (1) & (2) we can write,

​ = -​ +  = 0​​