Express \(r = (12, 210°)\) in the form \(a i + b j\).
Question 1 Report
Express \(r = (12, 210°)\) in the form \(a i + b j\).
Answer Details
We can express a vector in terms of its rectangular or Cartesian coordinates by using the formulas:
\[x = r\cos\theta\]
\[y = r\sin\theta\]
where r is the magnitude of the vector and θ is the angle it makes with the positive x-axis.
Using these formulas, we have:
\[x = r\cos\theta = 12\cos 210° = 12 \times \frac{-\sqrt{3}}{2} = -6\sqrt{3}\]
\[y = r\sin\theta = 12\sin 210° = 12 \times \frac{-1}{2} = -6\]
Therefore, we can express vector r as:
\[r = -6\sqrt{3}i - 6j = 6(-\sqrt{3}i - j)\]
Hence, the correct option is: \(6(-\sqrt{3}i - j)\).
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