Question

If 2î + 4ĵ - 5k̂ and î + 2ĵ + 3k̂ are two different sides of rhombus. Find the length of diagonals.
1. 7, \(\sqrt{69}\)
2. 6, \(\sqrt{59}\)
3. 5, \(\sqrt{65}\)
4. 8, \(\sqrt{45}\)

Answer 1

Correct Answer - Option 1 : 7, \(\sqrt{69}\)

Concept:

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

Suppose \(\vec{A} = aî + bĵ + ck̂\) and B = \(\vec{B}=pî + qĵ + rk̂\)

Then anyone diagonal of the rhombus is given by \(D_1=\vec{A}+\vec{B}\)

The other diagonal is given by \(D_2=\vec{B}-\vec{A}\)

The magnitude of the vector  \(A=\sqrt{a^2+b^2+c^2}\)

Calculation:

Given:

2î + 4ĵ - 5k̂ and î + 2ĵ + 3k̂ are 2 different sides of rhombus.

Suppose, \(\vec{A} = 2î + 4ĵ - 5k̂\) and B = \(\vec{B}=1î + 2ĵ + 3k̂\)

Then anyone diagonal of the rhombus is given by \(D_1=\vec{A}+\vec{B}\)

\(D_1=\vec{A}+\vec{B}\)

D₁ = (2î + 4ĵ - 5k̂) + (1î + 2ĵ + 3k̂)

D₁ = 3î + 6ĵ - 2k̂

The magnitude of the vector diagonal D₁

_{\(D_1=\sqrt{3^2+6^2+({-2})^2}\)}

D₁ = 7 

The other diagonal is given by \(D_2=\vec{B}-\vec{A}\)

\(D_2=\vec{B}-\vec{A}\)

D2 = (1î + 2ĵ + 3k̂) - (2î + 4ĵ - 5k̂)

D2 = - 1î - 2ĵ + 8k̂

The magnitude of the vector diagonal D₂

\(D_1=\sqrt{({-1})^2+({-2})^2+8^2}\)

\(D_1 =\sqrt{69} \)

∴ The length of diagonals of a rhombus is 7 and \(\sqrt{69} \).