If 𝐚 and 𝐛 are vectors in space given by 𝐚 = î + ĵ√5 and 𝐛 = 2î + ĵ + 3k√14, then the value of (2𝐚 + 𝐛).[(𝐚 × 𝐛) × (𝐚 − 𝐛)] is

Let the given expression be E. We have

E = (2𝐚 + 𝐛).[(𝐚 × 𝐛) × (𝐚 − 𝐛)]

We use the vector triple product identity:

p × (q × r) = (p.r)q − (p.q)r

Then (q × r) × p = −(p.r)q + (p.q)r

Therefore,

(𝐚 × 𝐛) × (𝐚 − 𝐛) = −[(𝐚 − 𝐛).𝐛]𝐚 + [(𝐚 − 𝐛).𝐚]𝐛

= −(𝐚.𝐛 − |𝐛|²)𝐚 + (|𝐚|² − 𝐚.𝐛)𝐛

Then

E = (2𝐚 + 𝐛).[−(𝐚.𝐛 − |𝐛|²)𝐚 + (|𝐚|² − 𝐚.𝐛)𝐛]

= −2(𝐚.𝐛 − |𝐛|²)|𝐚|² + (|𝐚|² − 𝐚.𝐛)(2𝐚.𝐛 + |𝐛|²)

Now, 𝐚 = î + ĵ√5, so |𝐚|² = 1 + 5 = 6

𝐛 = 2î + ĵ + 3k√14, so |𝐛|² = 4 + 1 + 9 = 14

𝐚.𝐛 = 2 + 1 = 3

Therefore,

E = −2(3 − 14)(6) + (6 − 3)(2(3) + 14)

= −2(−11)(6) + 3(6 + 14)

= 132 + 60 = 192