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

Concept:

Vectors and Coplanar Vectors:

• Two vectors and are said to be coplanar if there exists a scalar such that .
• Cross Product: The cross product of two vectors gives a vector that is perpendicular to both and , and its magnitude is , where is the angle between them.
• Condition for Coplanarity: If vectors are coplanar, then .

Calculation:

Given,

• We are asked to find the value of , where , , and .

For the vectors to be coplanar, their scalar triple product must be zero:

Using the definitions of :

Expanding the determinant gives:

Thus, we have or .

Since α and β are distinct, we conclude that .

Conclusion:

Hence, the value of is .

Calculation

Given:

Expanding the cross product:

Since , , and :

Now, taking the dot product with :

Using the scalar triple product identity and the fact that and :

Since and :

Given that this equals 54:

Hence, [a b c] = 3

Hence option 3 is correct

Calculation

For maxima ;

a = ±

for a = -

Hence option 3 is correct

Concept:

1. The vectors which are parallel to the same plane , or lie on the same plane are called coplanar vectors.

2. Three vectors are coplanar if their scalar triple product is zero. 

Calculation

Let  = 2i – j + k

 =i + 2j – 3k

= 3i + a j + 5k

⇒

⇒

⇒ 

⇒ 2(10 + 3a) - (-14) + (a - 6)

⇒ 20 + 6a + 14 + a - 6

⇒ 7a + 28

Since 7a + 28 = 0

⇒ a = -4

Hence Option(4) is correct.

CONCEPT:

Properties of Scalar Triple Product

• [a b c] = [b c a] = [c a b]
• [a b c] = - [b a c] = - [c b a] = - [a c b]
• [(a + b) c d] = [a c d] + [b c d]
• [λa, b c] = λ [a b c]
• Three non-zero vectors  are coplanar if and only if [a b c] = 0

CALCULATION:

Given:  are coplanar i.e 

⇒  and 

⇒ 

As we know that, [a b c] = [b c a] = [c a b]

⇒ 

As we know that, [λa, b c] = λ [a b c]

⇒ 

As we know that, vectors  are coplanar if and only if [a b c] = 0

⇒ 

⇒ 

Hence, correct option is  3.

CONCEPT:

• The scalar triple product of three vectors is zero if any two of them are equal
• If  is perpendicular to the vectors  then 

CALCULATION:

Given:  , where 

Statement 1:  is perpendicular to 

First let's find out 

⇒ 

As we know that, 

⇒ 

∵ It is given that 

⇒ 

⇒ 

Hence, statement 1 is true.

Statement 2:  is perpendicular to 

First let's find out 

⇒ 

As we know that, the scalar triple product of three vectors is zero if any two of them are equal

⇒ 

Hence, statement 2 is true.

Hence, the correct option is 3.

Concept:

Scalar Triple Product: 

If , 

 and,

Then their scalar triple product is defined as,

Calculation:

Given:

We have,

Now, 

If  is a unit vector so, 

When, θ = 0°

Maximum value = 

Concept:

If  are vectors, then

• For dot product 
• For cross product 

Calculation:

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

∴ 

Concept used:

a, b, and c vectors are coplanar when 

Calculation:

∴ λ = - 2

Concept:

1. If  and  be three vectors such that   and  are co-planar.

 = 0

2.Vector  is perpendicular to both vector   and .

Calculation:

We have,

  =  (Let)

⇒  is ⊥ to both  and .

But  is ⊥ to   and .

So,  or  will be coplanar to vector  and .

∴ Statement (1) is correct.

For statement 2:

Statement 2 is only possible when the vector  is parallel or anti-parallel to vector , which is not possible as per the above explanation for statement 1. So, this statement 2 is incorrect.

So, the best answer will be option 1.

Concept:

Scalar Triple Product:

A scalar triple product is also called a box product.

It is evident that scalar triple product of vectors means the product of three vectors. It means taking the dot product of one of the vectors with the cross product of the remaining two. It is denoted as [a b c ] = a · [b × c]

The product is cyclic in nature ⇔ [ a b c ] = [ b c a ] = [ c a b ]

Properties of the scalar triple product:

In a scalar triple product, dot and cross can be interchanged without altering the order of occurrences of the vectors ⇔ a · [b × c] = [a × b] ∙ c

Three vectors are coplanar if and only if their Scalar Triple Product is zero.

The Scalar Triple Product of three vectors is zero if any two of them are parallel.

Calculation:

Alternate solution:

As we know, cross product of parallel vectors is zero.

The Scalar Triple Product of three vectors is zero if any two of them are parallel.

CONCEPT:

• If ,  and , then .
• If vectors are coplanar then 

​
CALCULATION:

Given: The vectors  are coplanar.

As we know that, if vectors are coplanar then  

⇒ 

⇒ 2(0 - 4p) -(-1)(-15 - 0) + 1(12 - 0) = 0

⇒ -8p - 15 + 12 = 0

⇒ -8p - 3 =0 

⇒ -8p = 3

⇒ p = -3/8

Hence, correct option is 3.

Concept:

Condition of coplanar vectors: The three vectors are coplanar if their scalar triple product is zero.

Let  and  be the coplanar vectors.

Condition for coplanarity: 

Calculation:

Given: vectors  and  lie on a plane

Vectors lie on the same plane so vectors are coplanar.

Therefore, 

Expanding R₁, we get

⇒ α[0 − γ] − α[β − γ] + γ[γ−0] = 0

⇒ −αγ – αβ + αγ + γ² = 0

⇒ γ² = αβ

∴ α, β, γ are in G.P.

Concept:

If the three vectors are coplanar then their scalar triple product is zero..

⇒

Calculations:

Given the origin (0, 0, 0) and the points P(2, 3, 4), Q(1, 2, 3) and R(x, y, z) are co-planer

⇒ 

⇒ 

⇒ 

Here, ,  and are co planer

The three vectors are coplanar if their scalar triple product is zero..

⇒

⇒

⇒ x(9 - 8) - y(6 - 4) + z (4 - 3) = 0

⇒ x - 2y + z = 0

Hence, if the origin and the points P(2, 3, 4), Q(1, 2, 3) and R(x, y, z) are co-planar then x - 2y + z = 0