91Ó°ÊÓ

(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. $$ \mathbf{v}=3 \mathbf{i}+4 \mathbf{j}, \quad \mathbf{w}=-6 \mathbf{i}-8 \mathbf{j} $$

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ \sqrt{3}-i $$

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r=4 $$

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ 1-\sqrt{3} i $$

(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. $$ \mathbf{v}=3 \mathbf{i}-4 \mathbf{j}, \quad \mathbf{w}=9 \mathbf{i}-12 \mathbf{j} $$

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ -3 i $$

(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. $$ \mathbf{v}=4 \mathbf{i}, \quad \mathbf{w}=\mathbf{j} $$

(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. $$ \mathbf{v}=\mathbf{i}, \quad \mathbf{w}=-3 \mathbf{j} $$

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ 4-4 i $$

Find \(a\) so that the vectors \(\mathbf{v}=\mathbf{i}-a \mathbf{j}\) and \(\mathbf{w}=2 \mathbf{i}+3 \mathbf{j}\) are orthogonal.