If 2i + 4j - 5k and i + 2j + 3k are two different sides of rhombus. Find the length of diagonals.a)7, √69b)6, √59c)5 √65d)8, √45Correct answer is option 'A'. Can you explain this answer?

Question

If 2i + 4j - 5k and i + 2j + 3k are two different sides of rhombus. Find the length of diagonals.a)7, &radic;69b)6, &radic;59c)5 &radic;65d)8, &radic;45Correct answer is option 'A'. Can you explain this answer?

Answer

If aî + bĵ + ck̂ and pî + qĵ + rk̂ are 2 different sides of the rhombus.

Suppose

= aî+bĵ+ck̂ and B =

= pî+qĵ+rk̂

Then anyone diagonal of the rhombus is given by

The other diagonal is given by

The magnitude of the vector

The magnitude of the vector diagonal D₁

D₁ = 7

The other diagonal is given by

D₂ = - 1î - 2ĵ + 8k̂

The magnitude of the vector diagonal D₂

∴ The length of diagonals of a rhombus is 7 and √69.

Answer

If aî + bĵ + ck̂ and pî + qĵ + rk̂ are 2 different sides of the rhombus.

Suppose

= aî+bĵ+ck̂ and B =

= pî+qĵ+rk̂

Then anyone diagonal of the rhombus is given by

The other diagonal is given by

The magnitude of the vector

The magnitude of the vector diagonal D₁

D₁ = 7

The other diagonal is given by

D₂ = - 1î - 2ĵ + 8k̂

The magnitude of the vector diagonal D₂

∴ The length of diagonals of a rhombus is 7 and √69.

Answer

To find the length of the diagonals of the rhombus, we need to find the magnitude of the vectors 2i+4j-5k and i+2j+3k.

The magnitude of a vector is given by the square root of the sum of the squares of its components.

For the first vector, the magnitude is:
√((2^2) + (4^2) + (-5^2)) = √(4 + 16 + 25) = √45

For the second vector, the magnitude is:
√((1^2) + (2^2) + (3^2)) = √(1 + 4 + 9) = √14

So the length of the diagonals of the rhombus are √45 and √14.

However, the answer choices provided do not match these values. Therefore, the given answer choices are incorrect.

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