letumsnemesislh
2022-07-02
Answered

The points (2, -1, -2), (1, 3, 12), and (4, 2, 3) lie on a unique plane. Where does this plane cross the z-axis?

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Point K is on line segment

asked 2022-05-28

Solve for x in

$\frac{\mathrm{sin}(30)\cdot \mathrm{sin}(60)\cdot \mathrm{sin}(x+10)}{\mathrm{sin}(80)\cdot \mathrm{sin}(40)\cdot \mathrm{sin}(x)}=1$

$\frac{\mathrm{sin}(30)\cdot \mathrm{sin}(60)\cdot \mathrm{sin}(x+10)}{\mathrm{sin}(80)\cdot \mathrm{sin}(40)\cdot \mathrm{sin}(x)}=1$

asked 2022-07-15

Let $\mathrm{\u25b3}ABC$ be a triangle of sides AB, AC and BC which has the following equations

$AB\equiv {a}_{1}x+{b}_{1}y+{c}_{1}=0\phantom{\rule{0ex}{0ex}}AC\equiv {a}_{2}x+{b}_{2}y+{c}_{2}=0\phantom{\rule{0ex}{0ex}}BC\equiv px+qy+r=0$

Then an angle $\mathrm{\angle}A$ is acute if

$\left|\begin{array}{cc}{a}_{1}& {b}_{1}\\ p& q\end{array}\right|\left|\begin{array}{cc}p& q\\ {a}_{2}& {b}_{2}\end{array}\right|({a}_{1}{a}_{2}+{b}_{1}{b}_{2})<0$

And the angle $\mathrm{\angle}A$ is obtuse if

$\left|\begin{array}{cc}{a}_{1}& {b}_{1}\\ p& q\end{array}\right|\left|\begin{array}{cc}p& q\\ {a}_{2}& {b}_{2}\end{array}\right|({a}_{1}{a}_{2}+{b}_{1}{b}_{2})>0$

But I absolutely have no idea how to prove this theorem.

$AB\equiv {a}_{1}x+{b}_{1}y+{c}_{1}=0\phantom{\rule{0ex}{0ex}}AC\equiv {a}_{2}x+{b}_{2}y+{c}_{2}=0\phantom{\rule{0ex}{0ex}}BC\equiv px+qy+r=0$

Then an angle $\mathrm{\angle}A$ is acute if

$\left|\begin{array}{cc}{a}_{1}& {b}_{1}\\ p& q\end{array}\right|\left|\begin{array}{cc}p& q\\ {a}_{2}& {b}_{2}\end{array}\right|({a}_{1}{a}_{2}+{b}_{1}{b}_{2})<0$

And the angle $\mathrm{\angle}A$ is obtuse if

$\left|\begin{array}{cc}{a}_{1}& {b}_{1}\\ p& q\end{array}\right|\left|\begin{array}{cc}p& q\\ {a}_{2}& {b}_{2}\end{array}\right|({a}_{1}{a}_{2}+{b}_{1}{b}_{2})>0$

But I absolutely have no idea how to prove this theorem.

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A brass doorknob has a density of 8.0 grams per cubic centimeter and a volume of 6.5 cubic centimeters. What is its mass?

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Find

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Find the area of the parallelogram with vertices A(-3, 0), B(-1 , 3), C(5, 2), and D(3, -1).