91影视

It is given that \(胃\) is the angle between nonzero vectors \(u\) and \(v\).

We have to prove that \(胃\) is acute if and only if \(u 路 v > 0\).

To prove \(胃\) is acute if and only if \(u 路 v > 0\). Let \(u=\left<u_{1},v_{1} \right>\) and \(v=\left<u_{2},v_{2} \right>\) and \(胃\) be the angle between vectors \(u\) and \(v\) is \(cos\theta =\frac{u\cdot v}{\left\|u \right\|\left\|v \right\|}\).

As it is given that \(胃\) is acute which means \(胃\) lies between \(\left (0,\frac{\pi }{2} \right )\).

So, \(cos\theta>0\) which is possible when \(u 路 v > 0\).

Hence proved.

To prove that \(胃\) is right if and only if \(u 路 v = 0\). Let \(u=\left<u_{1},v_{1} \right>\) and \(v=\left<u_{2},v_{2} \right>\) and \(胃\) be the angle between vectors \(u\) and \(v\) is \(cos\theta =\frac{u\cdot v}{\left\|u \right\|\left\|v \right\|}\).

So, when \(\theta =90^{\circ }\)

\(cos90^{\circ } =\frac{u\cdot v}{\left\|u \right\|\left\|v \right\|}\)

\(0=\frac{u\cdot v}{\left\|u \right\|\left\|v \right\|}\)

So, \(0=\frac{u\cdot v}{\left\|u \right\|\left\|v \right\|}\) if and only if \(u 路 v = 0\).

Hence proved.

To prove that \(胃\) is obtuse if and only if \(u 路 v < 0\). Let \(u=\left<u_{1},v_{1} \right>\) and \(v=\left<u_{2},v_{2} \right>\) and \(胃\) be the angle between vectors \(u\) and \(v\) is \(cos\theta =\frac{u\cdot v}{\left\|u \right\|\left\|v \right\|}\).

As it is given that \(胃\) is obtuse which means \(胃\) lies between \(\left (\frac{\pi }{2},\pi \right )\).

So, \(cos\theta<0\) which is possible when \(u 路 v < 0\).

Hence proved.