Let vector a = i - αj + βk, vector b = 3i + βj - αk and vector c = -ai -2j + k, Where α and β are integers. if vector a.b

Question

Let \(\vec a = \hat i - a\hat j +\beta \hat k\),  \(\vec b = 3 \hat i + \beta \hat j - \alpha \hat k\)  and \(\vec c\) = -a\(\hat i\) - 2\(\hat j\) + \(\hat k\), Where \(\alpha \) and \(\beta\) are integers. if \(\vec a. \vec b\) = -1 and \(\vec b. \vec c\) = 10, then (\(\vec a \times \vec b\)). \(\vec c\) is equal to .... .

Answer 1

Correct answer is 9.

\(\vec a\) = (1, - α,β)

\(\vec b\) = (3, β, -α)

\(\vec c\) = (-α, -2,1); α,β ∈ 1

\(\vec a\).\(\vec b\) = -1 \(\Rightarrow\) 3 - α,β = -1

\(\Rightarrow\) α,β = 2

1	2
2	1
-1	-2
-2	-1

\(\vec b. \vec c\) =10

\(\Rightarrow\) -3α - 2β - α = 10

 \(\Rightarrow\) 2α + β + 5 = 0

\(\therefore\) α = -2; β = -1

= 1(–1 + 4) – 2 (3 – 4) – 1(– 6 + 2) 

= 3 + 2 + 4 = 9