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Addition of Vectors

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Question

Let $\to a, \to b$ and $\to c$ be unit vectors such that $\to a + \to b - \to c = 0$ . If the area of triangle formed by vectors $\to a$ and $\to b$ is $A$ , then what is the value of $4 A^{2}$ ?

A

3

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B

9

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C

\frac{3}{4}

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D

\frac{9}{4}

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Solution

The correct option is C $\frac{3}{4}$
Given, $\to a + \to b = \to c$
Now vector $\to c$ is along the diagonal of the parallelogram which has adjacent side vectors $\to a$ and $\to b$ .
Since $\to c$ is also a unit vector, triangle formed by vectors $\to a$ and $\to b$ is an equilateral triangle, then
Area of triangle $A = \frac{\sqrt{3}}{4}$

$4 A^{2} = 4 {(\frac{\sqrt{3}}{4})}^{2} = \frac{3}{4}$

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