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Question Number 159606 by mr W last updated on 19/Nov/21

Commented by mr W last updated on 19/Nov/21

$Q 158964$
	Answered by mr W last updated on 19/Nov/21

	Commented by mr W last updated on 19/Nov/21

	$METHOD I : vector method$ $a, b, c, d a r e u n i t v e c t o r s$ $l e t p = O A, q = O B$ $O C = p (a + c)$ $O D = q (b + d)$ $C D = q (b + d) - p (a + c)$ $C E = E D = \frac{q}{2} (b + d) - \frac{p}{2} (a + c)$ $A E = A C + C E = p c + \frac{q}{2} (b + d) - \frac{p}{2} (a + c)$ $\Rightarrow A E = \frac{q}{2} (b + d) - \frac{p}{2} (a - c)$ $B E = B D - E D = q d - \frac{q}{2} (b + d) + \frac{p}{2} (a + c)$ $\Rightarrow B E = - \frac{q}{2} (b - d) + \frac{p}{2} (a + c)$ $A E \cdot B E = - \frac{q^{2}}{4} (b \cdot b - d \cdot d) + \frac{p q}{4} (a \cdot b - a \cdot d - b \cdot c + c \cdot d) + \frac{p q}{4} (a \cdot b + a \cdot d + b \cdot c + c \cdot d) - \frac{p^{2}}{4} (a \cdot a - c \cdot c)$ $A E \cdot B E = - \frac{q^{2}}{4} (1 - 1) + \frac{p q}{4} (a \cdot b - a \cdot d - b \cdot c + c \cdot d) + \frac{p q}{4} (a \cdot b + a \cdot d + b \cdot c + c \cdot d) - \frac{p^{2}}{4} (1 - 1)$ $A E \cdot B E = \frac{p q}{2} (a \cdot b + c \cdot d)$ $A E \cdot B E = \frac{p q}{2} [\cos θ + \cos (π - θ)]$ $A E \cdot B E = \frac{p q}{2} (\cos θ - \cos θ)$ $A E \cdot B E = 0$ $\Rightarrow A E ⊥ B E$ $\Rightarrow y e l l o w a n g l e i s 90 °$
	Commented by Tawa11 last updated on 28/Nov/21

	$Great sir$
	Answered by mr W last updated on 19/Nov/21

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