The value of non-zero scalars α and β for all vectors \( a\) and \( b\) , such that \( α(2 a- b )+β( a + 2 b)=8 b - a \) is:

Compare the coefficients of \( a \) and \( b \) (they can be any vectors): \( a \): \(2α + β = -1\) \( b \): \(-α + 2β = 8\) From the first, \(β = -1 - 2α\). Substitute: \(-α - 2 - 4α = 8 α = -2\), then \(β = 3\). Answer: (D) α = −2, β = 3

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Question 48

Mathematics Scalar and Vector Products Hard

The value of non-zero scalars α and β for all vectors \(\vec a\) and \(\vec b\), such that \(\rm \alpha\left(2\vec a-\vec b \right)+\beta\left(\vec a + 2\vec b\right)=8\vec b - \vec a \) is:

(A) α = 2, β = 1

(B) α = -2, β = -3

(C) α = 1, β = 3

(D) α = -2, β = 3

✓ Correct Answer: Option D Verified Solution

📘 Step-by-Step Solution & Detailed Explanation:

Compare the coefficients of \(\vec{a}\) and \(\vec{b}\) (they can be any vectors):

\(\vec{a}\): \(2\alpha + \beta = -1\)
\(\vec{b}\): \(-\alpha + 2\beta = 8\)

From the first, \(\beta = -1 - 2\alpha\). Substitute: \(-\alpha - 2 - 4\alpha = 8 \Rightarrow \alpha = -2\), then \(\beta = 3\).

Answer: (D) α = −2, β = 3

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Question 48

📘 Step-by-Step Solution & Detailed Explanation:

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