For each pair of vectors u and v with angle θ between them, sketch the resultant.

|u| = 12, |v| = 20, θ = 27°

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Identify the given information: the magnitudes of vectors \( \mathbf{u} \) and \( \mathbf{v} \) are \( |\mathbf{u}| = 12 \) and \( |\mathbf{v}| = 20 \), and the angle between them is \( \theta = 27^\circ \).

Recall that the resultant vector \( \mathbf{R} = \mathbf{u} + \mathbf{v} \) can be found using the Law of Cosines for vectors, where the magnitude of \( \mathbf{R} \) is given by: \[ |\mathbf{R}| = \sqrt{|\mathbf{u}|^2 + |\mathbf{v}|^2 + 2 |\mathbf{u}| |\mathbf{v}| \cos(\theta)} \]

To sketch the resultant, start by drawing vector \( \mathbf{u} \) with length proportional to 12 units in any direction you choose.

Next, from the tip of \( \mathbf{u} \), draw vector \( \mathbf{v} \) at an angle of \( 27^\circ \) relative to \( \mathbf{u} \), with length proportional to 20 units.

The resultant vector \( \mathbf{R} \) is then drawn from the tail of \( \mathbf{u} \) (the starting point) to the tip of \( \mathbf{v} \) (the ending point). This vector represents the sum \( \mathbf{u} + \mathbf{v} \).

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