Determine whether the statement is true or false. If it is false, explain why. -If v = ai + bj = 0, then a = -b.

When u × v = 0 and u·v = 0, what can you conclude about u and v?

Find the coordinates of the point. -The point is located seven units in front of the yz-plane, two units to the left of the xz-plane, and one unit below the xy-plane.

Find (a) 2/3u, (b) 3v (c) v - u, and (d) 2u + 5v. u = ⟨4, 9⟩, v = ⟨2, -5⟩

Let $\mathbf{r}=\langle x, y, z\rangle$ and $\mathbf{r}_{0}=\langle 1,1,1\rangle$. Describe the set of all points (x, y, z) such that $\left\|\mathbf{r}-\mathbf{r}_{0}\right\|=2$.

Verify that the two planes are parallel and find the distance between the planes.

$$\begin{array}{l} 2 x+7 y+z=13 \\ 2 x+7 y+z=9 \end{array}$$

Three vertices of a parallelogram are (1, 2), (3, 1), and (8, 4). Find the three possible fourth vertices.

Find the unit vector in the direction of v and verify that it has length 1. v = ⟨3, 12⟩

Find the component form of u+v given the lengths of u and v and the angles that u and v make with the positive x-axis. ǁuǁ = 2, θ_u = 4 ǁvǁ = 1, θ_v = 2

Find the component form of u+v given the lengths of u and v and the angles that u and v make with the positive x-axis. ǁuǁ = 1, θ_u = 0° ǁvǁ = 3, θ_v = 45°

Find (a) u × v, (b) v × u, and (c) v × v. u = 3i + 5k v = 2i + 3j - 2k

Forces with magnitudes of 180 newtons and 275 newtons act on a hook. The angle between the two forces is $\theta$ degrees.

Use a graphing utility to graph the two functions $M$ and $\alpha$. Explain why one of the functions decreases for increasing values of $\theta$, whereas the other does not.

Use a computer algebra system or graphing utility to convert the coordinates of a point from one system to another among the rectangular, cylindrical, and spherical coordinate systems.

Determine whether the statement is true or false. If it is false, explain why or give an example. If u = ai + bj is a unit vector, then a² + b² = 1.