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If a(1) and a(2) are two non- collinear ...

If `a_(1) and a_(2)` are two non- collinear unit vectors and if `|a_(1)+a_(2)|=sqrt(3),` ,then value of `(a_(1)-a_(2)).(2a_(1)-a_(2))` is

A

2

B

`(3)/(2)`

C

`(1)/(2)`

D

1

Text Solution

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The correct Answer is:
To solve the problem, we need to find the value of \((\mathbf{a_1} - \mathbf{a_2}) \cdot (2\mathbf{a_1} - \mathbf{a_2})\) given that \(|\mathbf{a_1} + \mathbf{a_2}| = \sqrt{3}\) and that \(\mathbf{a_1}\) and \(\mathbf{a_2}\) are unit vectors. ### Step 1: Understand the given information We know: - \(|\mathbf{a_1}| = 1\) - \(|\mathbf{a_2}| = 1\) - \(|\mathbf{a_1} + \mathbf{a_2}| = \sqrt{3}\) ### Step 2: Use the formula for the magnitude of a vector sum Using the formula for the magnitude of the sum of two vectors: \[ |\mathbf{a_1} + \mathbf{a_2}|^2 = |\mathbf{a_1}|^2 + |\mathbf{a_2}|^2 + 2 \mathbf{a_1} \cdot \mathbf{a_2} \] Substituting the known values: \[ (\sqrt{3})^2 = 1^2 + 1^2 + 2 \mathbf{a_1} \cdot \mathbf{a_2} \] This simplifies to: \[ 3 = 1 + 1 + 2 \mathbf{a_1} \cdot \mathbf{a_2} \] \[ 3 = 2 + 2 \mathbf{a_1} \cdot \mathbf{a_2} \] \[ 2 \mathbf{a_1} \cdot \mathbf{a_2} = 1 \quad \Rightarrow \quad \mathbf{a_1} \cdot \mathbf{a_2} = \frac{1}{2} \] ### Step 3: Calculate the dot product \((\mathbf{a_1} - \mathbf{a_2}) \cdot (2\mathbf{a_1} - \mathbf{a_2})\) We can expand this dot product: \[ (\mathbf{a_1} - \mathbf{a_2}) \cdot (2\mathbf{a_1} - \mathbf{a_2}) = \mathbf{a_1} \cdot (2\mathbf{a_1}) - \mathbf{a_1} \cdot \mathbf{a_2} - \mathbf{a_2} \cdot (2\mathbf{a_1}) + \mathbf{a_2} \cdot \mathbf{a_2} \] This simplifies to: \[ = 2 \mathbf{a_1} \cdot \mathbf{a_1} - \mathbf{a_1} \cdot \mathbf{a_2} - 2 \mathbf{a_2} \cdot \mathbf{a_1} + \mathbf{a_2} \cdot \mathbf{a_2} \] Substituting the known values: \[ = 2(1) - \frac{1}{2} - 2\left(\frac{1}{2}\right) + 1 \] \[ = 2 - \frac{1}{2} - 1 + 1 \] \[ = 2 - \frac{1}{2} = 1.5 = \frac{3}{2} \] ### Final Answer Thus, the value of \((\mathbf{a_1} - \mathbf{a_2}) \cdot (2\mathbf{a_1} - \mathbf{a_2})\) is \(\frac{3}{2}\).
To solve the problem, we need to find the value of \((\mathbf{a_1} - \mathbf{a_2}) \cdot (2\mathbf{a_1} - \mathbf{a_2})\) given that \(|\mathbf{a_1} + \mathbf{a_2}| = \sqrt{3}\) and that \(\mathbf{a_1}\) and \(\mathbf{a_2}\) are unit vectors. ### Step 1: Understand the given information We know: - \(|\mathbf{a_1}| = 1\) - \(|\mathbf{a_2}| = 1\) - \(|\mathbf{a_1} + \mathbf{a_2}| = \sqrt{3}\) ...
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