Magnitude and Directions of a Vector MCQ Quiz - Objective Question with Answer for Magnitude and Directions of a Vector - Download Free PDF

Concept:

Magnitude of vector  then magnitude of vector is given by 

Calculation:

Given: Let where 

⇒ 

As we know that, if  then 

⇒ |

⇒  

Hence, option 1 is correct.

Given:

The vector with end points (5, –5, 0) and (2, 3, 0) 

Concept:

Let  be the position vector of the points A and B respectively,

then the vector equation of the line passing through A and B is given by I  I = II

Calculation:

Let the position vector of points A (-1, 0, 2) and B (3, 4, 6) be  and  respectively.

Then, 

And, 

⇒  

⇒ I  I = II

⇒ I  I = 

⇒ I  I = ​√​73

∴ Correct answer is √​73.

To determine the magnitude of a vector with given endpoints:

Calculation:

Step 1: Subtract the coordinates

Step 2: Square the differencesStep 3: Sum the squares

⇒ 9 + 64 + 0 = 73

Step 4: Take the square rootFinal Answer:

Hence, The Correct Answer is Option 4.

Thus, angles are

Answer : 2

Solution :

Let a̅ be the vector joining A(-2, 0, 3) and B(1, 4, 2).

∴ 

= 

and b = 

∴ Projection = 

= 

=  

= 1

Concept:

Let  then magnitude of the vector of a = 

Calculation:

Let  =  p(2î - ĵ + 2k̂)

Given, 

⇒ 

⇒ 

⇒ 3p = 3

∴ p = 1

Concept:

If , then 

Calculation:

Given A =  and B = 

 = 

 = 

Now 

 = 9√2

Concept:

The magnitude of a vector  = a₁î + a₂ĵ + a₃k̂ is given as .

The magnitude of the sum of vectors  and  can also be calculated as .

The resultant of a set of vectors acting at a point is simply the algebraic sum of the vectors.

Calculation:

The resultant of the vectors  = 2î + ĵ + k̂ and ​​ = î + 2ĵ + k̂ is:

 = (2î + ĵ + k̂) + (î + 2ĵ + k̂) = 3î + 3ĵ + 2k̂

Now, .

Concept:

Length of the vector  from origin is 

Calculation:

Length of the vector k(2î -  ĵ -  2k̂) from origin is 

=  

=  

= 3k

Length is 6 units given

3k = 6

k = 6/3

k = 2

Hence option 2 is correct.

CONCEPT:

If  is a vector then magnitude of  is given by: 

CALCULATION:

Given: 

Here, we have to find the value of 

⇒ 

As we know that, if   then magnitude of  is given by: 

⇒ 

Hence, option D is the correct answer.

Concept:

The direction cosines of the vector aî + bĵ + ck̂ are given by α = , β =  and γ = .

Calculation:

For the given vector 7î + 4ĵ - 3k̂, a = 7, b = 4 and c = -3.

The direction cosines of the vector are:

α = , β =  and γ = 

⇒ α = , β =  and γ =  

∴ (α , β , γ ) = () or (​)

Concept:

Unit vector: a vector that has a magnitude of one.

• Let
• Magnitude of vector of a =
• Unit vector =

Calculation:

Given Vector is

Now calculate the magnitude of Vector A,

So,  is a unit vector.

Concept:

• The angle between two vectors   and  is given by, ​ 
• If  = a₁ + a₂ + a₃ and   = b1 + b2 + b3, then  

Calculation:

Given: The angle between the vectors 2x2 + 3x +  and  −2 + x2 is obtuse

The angle between the vectors 2x2 + 3x +  and  −2 + x2 is given by

⇒ 

⇒ 

Since θ is obtuse, 

⇒ cos θ

⇒ 3x² - 6x

⇒ x(x - 2)

⇒ 0

∴ The correct option is (1).

Concept:

If A and B are points with position vectors  respectively then 

If  is a vector then the magnitude of the vector is given by

Calculation:

Given: The position vectors of points A and B are  and  respectively

As we know, If A and B are points with position vectors  respectively then 

As we know that, If  is a vector then the magnitude of the vector is given by

Given:

Calculation:

We have,

⇒ 

⇒          [∵ ]

⇒ 

⇒        []

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Put the value of 

⇒ 

⇒  

∴   and