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

Concept:

If  are two vectors parallel to each other then  or 

Calculation:

Given:

  and  are parallel vectors,

Therefore, 

Equating the coefficient of 

⇒ 1 = 3λ, ∴ λ = 1/3            

⇒ -a = -6λ 

⇒ 5 = bλ                 .... (1)

Put the value of λ in equation (1), we get

5 = b × (1/3)

So, b = 15

Concept:

Equal Vectors
Two or more vectors are said to be equal when their magnitude is equal and also their direction is the same.

Calculation:

Given:  and  

∴ a₃ = 1

Concept:

Equal Vectors
Two or more vectors are said to be equal when their magnitude is equal and also their direction is the same.

Calculation:

Given:   and   

∴ a3 = - 2

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.

Concept:

Conditions of collinear vector:

• Three points with position vectors  are collinear if and only if the vectors  and  are parallel. ⇔ 
• If the points (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) be collinear then 

Solution:

We know that, If the points (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3) be collinear then 

Given  , ,  are collinear

∴ 

⇒ 1 (-20 + 14) – (2) (-5λ + 21) + 3 (-2λ + 12) = 0

⇒ -6 + 10λ – 42 - 6λ + 36  = 0

⇒ 4λ = 12

∴ λ = 3

Concept:

Three or more points are collinear, if slope of any two pairs of points is same.

Here, 

Let, A = (5, -2), B = (8, -3), C = (a, -12)

Now, slope of AB = Slope of BC = Slope of AC ....(∵ points are collinear)

⇒ a - 8= 27

⇒ a = 27 + 8 = 35

Hence, option (4) is correct.

Concept:

For two vectors  to be collinear,​  where λ is a scalar.

Calculation:

Given that, the vectors  &  are collinear.

Since two vectors  are collinear then  where λ is a scalar.

⇒ 

⇒ 

⇒ λp = 4,  λq = 1 and -2λ = -3

⇒  λ = 3/2 

So, by substituting λ = 3/2 in  λp = 4 and λq = 1, we get

⇒ (3/2)p = 4 and (3/2)q = 1

⇒ p = 8/3 and q  = 2/3

∴  is the correct answer.

Concept:

If  are two vectors parallel to each other then  or 

Calculation:

Given:

  and  are parallel vectors,

Therefore, 

Equating the coefficient of 

⇒ 1 = 3λ, ∴ λ = 1/3            

⇒ -a = -6λ 

⇒ 5 = bλ                 .... (1)

Put the value of λ in equation (1), we get

5 = b × (1/3)

So, b = 15

Concept:

If  are two vectors parallel to each other then  or 

Calculation:

Given:

  and  are parallel vectors,

Therefore, 

Equating the coefficient of 

⇒ x = 2λ                    .... (1)

⇒ -2 = -4λ 

∴ λ = 1/2

Put the value of λ in equation (1), we get

x = 2 × (1/2)

So, x = 1

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 used:

a, b, and c vectors are coplanar when 

Calculation:

⇒ 2(10 + 3a) + 1(5 + 9) + 1(a - 6) = 0

⇒ 20 + 6a + 14 + a - 6 = 0

⇒ 7a = - 28

∴ a = - 4

∴ The vector 2î - ĵ + k̂, î + 2ĵ - 3k̂ and 3î + aĵ + 5k̂ are coplanar when a = - 4

CONCEPT:

If vectors  are perpendicular then 

CALCULATION:

Given:  and  are perpendicular

∵  and  are perpendicular

⇒ λ + 9 = 0

⇒ λ = - 9

Hence, option A is the correct answer.

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

⇒