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Problem 617

AIME late

Algebra Difficulty 5.0 Find the answer

5. Let vectors $a$ and $b$ satisfy $|a|=1,|b|=2$ , and the angle between $a$ and $b$ is $60^{\circ}$ . If the angle between the vectors $7 a+2 t b$ and $t a+b$ is obtuse, then the range of the real number $t$ is $\qquad$

Your answer

Official solution

5. $-7<t<-\frac{1}{2}$ , and $t \neq-\frac{\sqrt{14}}{2}$ .

From $(7 a+2 t b) \cdot(t a+b)<0$ , we get $-7<t<-\frac{1}{2}$ .
Moreover, the angle between $7 a+2 t b$ and $t a+b$ must not be a straight angle, hence, $t \neq-\frac{\sqrt{14}}{2}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.