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The vectors $a = (1,1,1)$ and $b = (−1, −1, −1)$ are given. Determine the unit vector $c$ such that $\angle (a, c) = \frac \pi 6$ and that the area of ​​the parallelogram constructed over the vectors $b$ and $c$ is equal to $\sqrt2.$

So I tried solving this like:

$$P=|b \times c|=\sqrt2 \Rightarrow b \times c=(y-z, z-x, x-y)$$

$$|c|=1 \Rightarrow x^2 +y^2+c^2=1$$

and $$∡ (a, c) = π / 6 \Rightarrow ac=|a||c|\cdot \cos ∡ (a, c)$$

And with this I got 3 equations but i really don't know if this is correct or not. Can you help me solve this, I need it for my homework.

Sunshine
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    Vectors $a$ and $b$ seem to be aligned and for $c$ unitary the area for the paralellogram should be $3/2$. – user Sep 13 '21 at 22:20
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    Here's a MathJax tutorial :) – Shaun Sep 13 '21 at 22:20
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    I don't believe there are any solutions. – aschepler Sep 13 '21 at 22:21
  • I got three equations the first one (1) x^2+y^2+z^2=1 i got it from |c|=sqrt( x^2+y^2+z^2)=1

    (2) x+y+z=3/2 from a*c=|a||c|⋅cos∡(a,c) => (x+y+z)/sqrt(3)=sqrt(3)/2

    and (3) (y-z)^2+(z-x)^2+(x-y)^2=2 i got it from P=|bxc|=sqrt(2) bxc=(y-z,x-z,x-y)

    But it doesn't look me right, i think I made mistake somwhere so please if you can help me it would mean a lot to me. Thank you all.

     – Sunshine Sep 13 '21 at 22:30
  • @Sunshine Check the input data for the problem. – user Sep 13 '21 at 22:35
  • @user This question was even on exam few years ago. It cant be mistake in the input data. But thank you for trying to help me. – Sunshine Sep 13 '21 at 22:38

1 Answers1

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As currently stated, the problem has no solution, indeed vectors $a$ and $b$ are aligned with lenght $\sqrt 3$ therefore for any vactor $c$ such that $\angle (a, c) = \frac \pi 6$ we have that the area of the parallelogram is

$$A=\sqrt 3 \cdot 1 \sin\left(\frac \pi 6\right)= \frac 3 2\neq \sqrt 2$$

user
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  • But area of paralelogram is for vectors b and c not for a and c and their angle: P= ({\sqrt 3})1sin∠(b,c)={\sqrt 2} => sin∠(b,c)=sqrt(2)/sqrt(3) I understand that |a|=|b|= sqrt(3) but how are they aligned? – Sunshine Sep 13 '21 at 22:57
  • @Sunshine We have that $a=-b \iff a+b=0$ which means that they are aligned. – user Sep 13 '21 at 22:59
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    I am little disappointed, because it doesn't have solution. Anyway thank You a lot, You saved me sleepless nights. Sry for my bad English :) – Sunshine Sep 13 '21 at 23:06
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    @Sunshine You are welcome! Check for the errata corrige of the book. Bye – user Sep 13 '21 at 23:07