Projection of Vector on a line or direction MCQ Quiz - Objective Question with Answer for Projection of Vector on a line or direction - Download Free PDF

Calculation

Given:

Vertices: , , 

1) Find the lengths of AB and AC:

2) Let D be the point where the angle bisector of intersects BC.

By the angle bisector theorem,

3) Find the coordinates of D using the section formula:

4) Find the length of AD:

⇒

⇒

∴ The length of the internal bisector of is .

Hence option 2 is correct

Explanation:

The additive inverse vector of vector pi + 2j - 3k is

-pi - 2j + 3k

So,  = -pi - 2j + 3k

Comparing both sides

So, p = 9, b = 4

Putting in  we get

i.e., 4b = 91

i.e., b = 81/4

Hence p = 9, b = 4, a = 81/4

Option (2) is true.

Concept Used:

Projection of vector on is

Calculation

Given:

, , 

Projection of AB on AC =

Projection of AB on AC =

Hence option 2 is correct

Calculation

Let

Projection of on

or

For  

Hence option 3 is correct

Answer : 3

Solution :

Vector projection of  on  

=  

=  

= 

Concept:

projection of  in the direction of    = 

Calculations:

Given, the angle between  and the projection of  in the direction of  is -2

We know that, 

projection of  in the direction of = 

⇒ - 2 = 

⇒ - 2 = 

⇒ - 2 = 

⇒  

Hence, if the angle between  and the projection of  in the direction of  is -2, then  4

Concept:

The vector form of the component of  along  = 

Calculations:

Given,   and 

⇒ 

⇒ 

and  

The vector form of the component of  along  = 

= 

Hence, if  and , then the vector form of the component of  along  is 

Concept:

Projection of any vector   on vector  is:

P = 

Where  is unit vector in direction of vector 

 = 

Calculation:

Given  = i + 2j + k and  = -i + j - 3k

Projection of  on  (let P)= 

Now, unit vector in direction of vector  is 

(-i + j - 3k)

Now P = (i + 2j + k) ⋅ (-i + j - 3k)

P = (-1 + 2 - 3)

P = 

Explanation:

The projection vector of  on  is given by .

The correct answer is option 2.

Concept:

The projection of vector   on  is given by .

Calculation:

Given   = 2î − ĵ + k̂  and   = î + 2ĵ + 2k̂

  = ( 2î − ĵ + k̂ )⋅ ( î + 2ĵ + 2k̂ )

= 2 × 1 + (-1) × 2 + 1 × 2 

= 2 - 2 + 2

= 2

|| = | î + 2ĵ + 2k̂ |

= 

=  

= 3

The projection of vector   on   = \(\frac{\rm\vec{a}⋅\rm\vec{b}}{|\vec b|}\).

= 

The projection of vector  = 2î − ĵ + k̂ along  = î + 2ĵ + 2k̂ is 2/3.

The correct answer is option 1.

Concept:

Let  be two vectors. Then the vector component of the vector  is given by:

Calculation:

Given:

 and 

|| = √52 

The component of vector a along b is 

Mistake PointsWe are asked to find the vector component of 'a' along 'b'. If the scalar component was asked, then there would have been only 5 in the denominator.

Concept:

projection of  in the direction of    = 

Calculations:

Given, the angle between  and the projection of  in the direction of  is -2

We know that, 

projection of  in the direction of = 

⇒ - 2 = 

⇒ - 2 = 

⇒ - 2 = 

⇒  

Hence, if the angle between  and the projection of  in the direction of  is -2, then  4

Concept:

Referring to the diagram for this given data, the projection of  on  can be calculated as,

= 

Calculation:

Given:

P(1, -1, 3), Q(2, -4, 11), A(-1, 2, 3), B(3, -2, 10) 

.

 = 8

So, the correct answer will be option 3. 

Concept:

If vector v = ai + bj + ck, then magnitude of vector v is .

Projection of vector v on c is 

A vector  in the plane of  and  is 

Calculation:

A vector  in the plane of  and  is  

⇒ 

⇒  = 

Projection of  on (given)

⇒ 

⇒             [ = √3]

⇒ 

⇒ 

∴ Vector 

Concept:

Let  be two vectors. Then the scalar projection of the vector  is given by: 

Calculation:

Now, by using the cosine formula,

Projection of  on  ,