91Ó°ÊÓ

Draw the angle \(\frac{5 \pi}{6}\) in standard position.

Is the sine function even, odd, or neither?

True or False Given two nonzero, nonorthogonal vectors \(\mathbf{v}\) and \(\mathbf{w},\) it is always possible to decompose \(\mathbf{v}\) into two vectors, one parallel to \(\mathbf{w}\) and the other orthogonal to \(\mathbf{w}\).

Multiple Choice In a rectangular coordinate system, where does the point with polar coordinates \(\left(1,-\frac{\pi}{2}\right)\) lie? (a) in quadrant IV (b) on the \(y\) -axis (c) in quadrant II (d) on the \(x\) -axis

If \(\mathrm{v}\) is a vector with initial point \(\left(x_{1}, y_{1}\right)\) and terminal point \(\left(x_{2}, y_{2}\right),\) then which of the following is the position vector that equals \(\mathbf{v} ?\) (a) \(\left\langle x_{2}-x_{1}, y_{2}-y_{1}\right\rangle\) (b) \(\left\langle x_{1}-x_{2}, y_{1}-y_{2}\right\rangle\) (c) \(\left\langle\frac{x_{2}-x_{1}}{2}, \frac{y_{2}-y_{1}}{2}\right\rangle\) (d) \(\left\langle\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right\rangle\)

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=\mathbf{i}+\mathbf{j}, \quad \mathbf{w}=-\mathbf{i}+\mathbf{j} $$

True or False A cardioid passes through the pole.

In Problems \(13-24,\) plot each complex number in the complex plane and write it in polar form and in exponential form. $$ 1+i $$

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=3 \mathbf{i}+4 \mathbf{j}, \quad \mathbf{w}=-6 \mathbf{i}-8 \mathbf{j} $$

Find \(b\) so that the vectors \(\mathbf{v}=\mathbf{i}+\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+b \mathbf{j}\) are orthogonal.