Given that w = 8i + 3j, x = 6i - 5j, y = 2i + 3j and |z| = 41. find z in the direction of w + x - 2y.

Assessment: WAEC SSCE - Further Mathematics - 2020 (Essay) Subject: Further Mathematics

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First find the direction vector \(w + x - 2y\):

\[w + x - 2y = (8i + 3j) + (6i - 5j) - 2(2i + 3j)\] \[= (8 + 6 - 4)i + (3 - 5 - 6)j = 10i - 10j\]

Its magnitude is:

\[|10i - 10j| = \sqrt{10^2 + (-10)^2} = \sqrt{200} = 10\sqrt{2}\]

The unit vector in this direction is:

\[\hat{u} = \frac{10i - 10j}{10\sqrt{2}} = \frac{1}{\sqrt{2}}i - \frac{1}{\sqrt{2}}j\]

Since \(|z| = 41\) and \(z\) points in this direction:

\[z = 41\hat{u} = \frac{41}{\sqrt{2}}i - \frac{41}{\sqrt{2}}j = \frac{41\sqrt{2}}{2}(i - j)\]

Therefore \(z \approx 29.0i - 29.0j\).

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