Scalar Triple Product MCQ Quiz in मल्याळम - Objective Question with Answer for Scalar Triple Product - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Concept:

Scalar Triple Product: 

If ,  and  then their scalar triple product is defined as 

Calculation:

Given: 

As we know that, the scalar triple product of three vectors is 

⇒ 

⇒ 

Hence, the correct option is 2.

Concept:

Scalar Triple Product: 

If ,  and  then their scalar triple product is defined as 

Calculation:

Given: 

As we know that, the scalar triple product of three vectors is 

⇒ 

⇒ 

Hence, the correct option is 4.

Concept:

The scalar triple product of three vectors  and  is denoted by

where, ,  and 

Calculation:

We have,  and  are three non-coplanar vectors,

∴ 

∴ 

= 

= 

= 1 + 1 - 1

= 1

∴ The value of  is 1.

The correct answer is Option 3.

Concept:

If ,  and , then .

Note: 

Calculation:

Let  ,  and ,

As we know that, if 

Hence, option 2 is correct.

Concept:

Scalar Triple Product: 

If ,  and  then their scalar triple product is defined as

Calculation:

Given: 

As we know that, the scalar triple product of three vectors is 

⇒ 

Hence, the correct option is 3.

Concept:

 = 

Calculations:

Given, 

then  = 

⇒ = [1(1 + x - y) - x( 1 - x)] - 0 - 1(x² - y)

⇒ = 1 + x - y - x + x²  - x2 + y

⇒ = 1

Hence,   depends on neither on x nor on y.

Concept:

For two vectors  and  at an angle θ to each other:

• Dot Product is defined as .
• Cross Product is defined as  where  is the unit vector perpendicular to the plane containing  and .

For three vectors ,  and :

• Triple Cross Product: is defined as: .
• Triple Scalar Product (Box Product): is defined as .

Volume of a parallelepiped, with vectors ,  and  as its sides, is given by the box product of the three vectors.

• Volume = .

Calculation:

The sides of the parallelepiped are:

= 2î + 3ĵ + 4k̂

= î + αĵ + 2k̂

= î + 2ĵ + αk̂

∴ Volume =  = 15

⇒ 2(α² - 4) + 3(2 - α) + 4(2 - α) = 15

⇒ 2α2 - 8 + 6 - 3α + 8 - 4α = 15

⇒ 2α2 - 7α - 9 = 0

⇒ 2α2 - 9α + 2α - 9 = 0

⇒ α(2α - 9) + (2α - 9) = 0

⇒ (2α - 9)(α + 1) = 0

⇒ 2α - 9 = 0 OR α + 1 = 0

⇒ α =  OR α = -1.

Concept:

Scalar Triple Product: 

If ,  and  then their scalar triple product is defined as 

Calculation:

Given: 

As we know that, the scalar triple product of three vectors is 

⇒ 

⇒ 

Hence, the correct option is 1.

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:

If  are three vectors, then  is called the scalar triple product or box product, denoted by [abc]

• [abc] = [bca] = [cab] (It follows cyclic order)

For four vectors,  quadruple product is given by:

 = [abd]c - [abc]d

Calculation:

We have, 

[pqr] = 

      ...........(1)

∵ 

      .......(2)

put this in (1), 

The correct answer is Option 1