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Find unit vector making an obtuse angle with x-axis and perpendicular to the plane containing the points $A(1,2,3), B(2,3,4), C(1,5,7)$.

I managed to derive the plane containing the points $A, B, C$ viz. $x-4y+3z-2=0$ but cannot proceed from here.

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Hint: for any plane given by the equation $ax + by + cz + d = 0$, the vector $(a, b, c)$ originating from the origin is perpendicular to the plane. So in your case $(1, -4, 3)$ is perpendicular to your plane. Now, it remains for you to answer these questions

  • How do you modify that vector to get a unit vector?
  • If the unit vector you got above is not obtuse with the $x$-axis, how do further modify it so that the new unit vector is? In fact, how do you determine the angles unit vectors make with the $x$-axis?
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  • $\begingroup$ Divide it by ✓26 $\endgroup$ – Samar Imam Zaidi Apr 7 at 11:52
  • $\begingroup$ Yes, that will give you your unit vector. Now, how do you calculate the angle it makes with the $x$-axis? $\endgroup$ – ZeroXLR Apr 7 at 11:54
  • $\begingroup$ Dot product should be less than zero $\endgroup$ – Samar Imam Zaidi Apr 7 at 11:57
  • $\begingroup$ Hence <-1,4,-3> is the direction so that <1,0,0> & <-1,4,-3> dot product is negative obtusr $\endgroup$ – Samar Imam Zaidi Apr 7 at 11:59
  • $\begingroup$ Yes, but don't forget to divide by $\sqrt{26}$ i.e. $(-1, 4, -3) / \sqrt{26}$ is the final answer. $\endgroup$ – ZeroXLR Apr 7 at 12:02

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