Understanding the Problem
The question asks us to find the value of the expression \(x^2 + y^2 + z^2\), given the definitions of \(x\), \(y\), and \(z\) in terms of \(a\), \(b\), and the cube roots of unity (\(1, \omega, \omega^2\)).
Key Properties of Cube Roots of Unity
Before we start calculating, let's recall the essential properties of the cube roots of unity:
- The cube roots of unity are \(1\), \(\omega\), and \(\omega^2\).
- They satisfy the equation \(t^3 = 1\).
- The sum of the cube roots of unity is zero: \(1 + \omega + \omega^2 = 0\).
- Also, \(\omega^3 = 1\).
- From \(1 + \omega + \omega^2 = 0\), we can derive other relations like \(1 + \omega = -\omega^2\), \(1 + \omega^2 = -\omega\), and \(\omega + \omega^2 = -1\).
- Also \(\omega^4 = \omega^3 \cdot \omega = 1 \cdot \omega = \omega\).
Given Expressions
We are given:
- \(x = a + b\)
- \(y = a\omega + b\omega^2\)
- \(z = a\omega^2 + b\omega\)
Calculating the Squares
Now, let's calculate the square of each expression:
-
Calculating \(x^2\):
Using the formula \((p+q)^2 = p^2 + 2pq + q^2\): \(x^2 = (a + b)^2 = a^2 + 2ab + b^2\)
-
Calculating \(y^2\):
Using the same formula: \(y^2 = (a\omega + b\omega^2)^2\) \(y^2 = (a\omega)^2 + 2(a\omega)(b\omega^2) + (b\omega^2)^2\) \(y^2 = a^2\omega^2 + 2ab\omega^3 + b^2\omega^4\) Using the properties \(\omega^3 = 1\) and \(\omega^4 = \omega\): \(y^2 = a^2\omega^2 + 2ab(1) + b^2\omega\) \(y^2 = a^2\omega^2 + 2ab + b^2\omega\)
-
Calculating \(z^2\):
Similarly: \(z^2 = (a\omega^2 + b\omega)^2\) \(z^2 = (a\omega^2)^2 + 2(a\omega^2)(b\omega) + (b\omega)^2\) \(z^2 = a^2\omega^4 + 2ab\omega^3 + b^2\omega^2\) Using the properties \(\omega^3 = 1\) and \(\omega^4 = \omega\): \(z^2 = a^2\omega + 2ab(1) + b^2\omega^2\) \(z^2 = a^2\omega + 2ab + b^2\omega^2\)
Summing the Squares
Now, we need to find the sum \(x^2 + y^2 + z^2\): \(x^2 + y^2 + z^2 = (a^2 + 2ab + b^2) + (a^2\omega^2 + 2ab + b^2\omega) + (a^2\omega + 2ab + b^2\omega^2)\)
Let's group the terms involving \(a^2\), \(ab\), and \(b^2\): \(x^2 + y^2 + z^2 = (a^2 + a^2\omega^2 + a^2\omega) + (2ab + 2ab + 2ab) + (b^2 + b^2\omega + b^2\omega^2)\)
Factor out \(a^2\), \(2ab\), and \(b^2\): \(x^2 + y^2 + z^2 = a^2(1 + \omega^2 + \omega) + 2ab(1 + 1 + 1) + b^2(1 + \omega + \omega^2)\)
Applying the Properties
Using the property \(1 + \omega + \omega^2 = 0\): \(x^2 + y^2 + z^2 = a^2(0) + 2ab(3) + b^2(0)\)
Simplifying the expression: \(x^2 + y^2 + z^2 = 0 + 6ab + 0\) \(x^2 + y^2 + z^2 = 6ab\)
Conclusion
Therefore, the value of \(x^2 + y^2 + z^2\) is \(6ab\). This matches option 1.
Similar Questions
If
\(\left( \frac{1-i}{1+i} \right)^{2m} \left( \frac{1+i}{1-i} \right)^{2n} = 1\)
where \(i = \sqrt{-1}\), then what is the smallest positive value of \((m – n)\)?
- If \(z\) is any complex number and \(iz^3 + z^2 - z + i = 0\), where \(i = \sqrt{-1}\), then what is the value of \((|z|+1)^2\)?
- Let \(z_1\) and \(z_2\) be two complex numbers such that \(\left|\frac{z_1+z_2}{z_1-z_2}\right| = 1\), then what is \(\text{Re}\left(\frac{z_1}{z_2}\right)+1\) equal to ?
- If \(\omega \ne 1\) is a cube root of unity, then what are the solutions of \((z-100)^3 + 1000 = 0\) ?
- What is \((1+i)^4 + (1-i)^4\) equal to, where \(i = \sqrt{-1}\)?
- If \(\omega \neq 1\) is a cube root of unity, then what is \((1 + \omega - \omega^2)^{100} + (1 - \omega + \omega^2)^{100}\) equal to?
- What is the value of the sum
\(\sum_{n=1}^{20}(i^{n-1} + i^n + i^{n+1})\)
where \(i = \sqrt{-1}\)? - If \(x, y\) and \(z\) are the cube roots of unity, then what is the value of \(xy + yz + zx\)?
If
\(\alpha = \frac{-1+\sqrt{-3}}{2}\)
then what is the value of
\((1 + \alpha^{19} – \alpha^{35})^{100} – (1 – 3\alpha^{25} + \alpha^{38})^{50}\)?
- If \(\alpha\), \(\beta\), \(\gamma\) are cube roots of -8, then what is \(\frac{\alpha^2 p^2 + \beta^2 q^2 + \gamma^2 r^2}{\beta^2 p^2 + \gamma^2 q^2 + \alpha^2 r^2}\) equal to ?
Important Questions from Complex Numbers
If \(x + iy = \sqrt {\frac{{a + ib}}{{c + id}}}\), then the value of x2 + y2 is -
If x = cos θ + i sin θ, then the value of \({x^n} + \frac{1}{{{x^n}}}\) is:
If \(\left| {\begin{array}{*{20}{c}} {6i}&{ - 3i}&1\\ 4&{3i}&{ - 1}\\ {20}&3&i \end{array}} \right| = x + iy\), then the values of x and y are:
If iz3 + z2 - z + i = 0, then the value of |z| is:
Nature of the triangle formed by the points representing the complex numbers 3 + 4i, 8 - 6i and 13 + 9i is:
