A vector with magnitude 10 points in a direction 235 degrees counterclockwise from the positive x axis Write the vector in component form.

To write the vector in component form, we need to use the magnitude and direction of the vector. The vector has a magnitude of 10 and points in a direction of 235 degrees counterclockwise from the positive x-axis.<br /><br />The component form of a vector \(\mathbf{v}\) with magnitude \(r\) and direction \(\theta\) is given by:<br /><br />\[<br />\mathbf{v} = \langle r \cos \theta, r \sin \theta \rangle<br />\]<br /><br />Here, \(r = 10\) and \(\theta = 235^\circ\).<br /><br />First, convert the angle from degrees to radians if necessary, but since most calculators can handle degrees directly, we'll proceed with degrees:<br /><br />\[<br />\mathbf{v} = \langle 10 \cos 235^\circ, 10 \sin 235^\circ \rangle<br />\]<br /><br />Now calculate the components:<br /><br />1. Calculate \(\cos 235^\circ\):<br /> - \(\cos 235^\circ = \cos(180^\circ + 55^\circ) = -\cos 55^\circ\)<br /><br />2. Calculate \(\sin 235^\circ\):<br /> - \(\sin 235^\circ = \sin(180^\circ + 55^\circ) = -\sin 55^\circ\)<br /><br />Using approximate values for \(\cos 55^\circ\) and \(\sin 55^\circ\):<br /><br />- \(\cos 55^\circ \approx 0.5736\)<br />- \(\sin 55^\circ \approx 0.8192\)<br /><br />Substitute these into the expressions:<br /><br />\[<br />\mathbf{v} = \langle 10 \times (-0.5736), 10 \times (-0.8192) \rangle<br />\]<br /><br />\[<br />\mathbf{v} = \langle -5.736, -8.192 \rangle<br />\]<br /><br />Therefore, the vector in component form is approximately \(\langle -5.736, -8.192 \rangle\).