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Convexity

If a = i + ...

Question

If $¯ ¯ ¯ a = ¯ i + 2 ¯ j, ¯ ¯ b = - 2 ¯ i + ¯ j, ¯ ¯ c = 4 ¯ i + 3 ¯ j$ , find $x$ and $y$ such that $¯ ¯ c = x ¯ ¯ ¯ a + y ¯ ¯ b$ .

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Solution

Given, $¯ ¯ c = x ¯ ¯ ¯ a + y ¯ ¯ b$
$4 ¯ i + 3 ¯ j = x (¯ i + 2 ¯ j) + y (- 2 ¯ i + ¯ j)$
$4 ¯ i + 3 ¯ j = (x - 2 y) ¯ i + (2 x + y) ¯ j$
Comparing both the sides, we get
$x - 2 y = 4$
and $2 x + y = 3$
Solving the equations, we get
$x = 2$ and $y = - 1$

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\to u = ˆ i + 2 ˆ j

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