Question

The area of a parallelogram formed by the vectors A=î+2j+3k and B=3î-2j+k as adjacent sides is Lambda√3. find the value of lambda.
Hint: Calculate A×B and area of parallelogram = |A×B|​

Answer 1

Given,

• A Parallelogram with adjacent sides to be two vectors, A = i + 2j + 3k and B = 3i - 2j + k.
• The area of the parallelogram is given to be λ√3 sq. units.

To Find,

• Value of λ

Solution,

Given two adjacent sides of a Parallelogram as two vectors A and B. Then, The area of the parallelogram is given by the cross product of the two adjacent sides i.e., A × B.

So,

| i j k |

⇒ A × B = | 1 2 3 |

| 3 -2 1 |

⇒ A × B = (2×1 - 3×-2)i - (1×1 - 3×3)j + (1×-2 - 3×2)k

⇒ A × B = (2 + 6)i - (1 - 9)j + (-2 - 6)k

⇒ A × B = 8i + 8j - 8k

Now, Let us find the area of the given parallelogram which is the magnitude of A × B i.e., | A × B |

⇒ Area = √( 8² + 8² + (-8)² )

⇒ Area = √( 3 × 8²)

⇒ Area = 8√3 sq. units

But, Given that area of ||gm is λ√3 sq. units, So

⇒ λ√3 = 8√3

⇒ λ = 8

Hence, The value of Lambda, λ is 8.

Answer 2

Given :

• The area of a parallelogram formed by the vectors A=î+2j+3k

• B=3î-2j+k as adjacent sides is Lambda√3

To Find:

• Find the value of lamba

Solution:

We know that,

For a parallelogram having adjacent sides â and b

According to the question :

A × B = (2×1 - 3×-2)i - (1×1 - 3×3)j + (1×-2 - 3×2)k

A × B = 8î - ( - 8 )j + ( - 8 )k

A × B = 8i + 8j - 8k

A × B i.e., | A × B |

Substitute all values :

Area = √( 8² + 8² + (-8)²

Area = √( 8² + 8² + (-8)² )

Area = √( 64 + 64 + 64 )

Area = √( 64 × 3 )

Area = 8√3

Lamba = 8

The value of Lambda is 8